RANDOMNESS OF EUROMILLIONS DRAWS
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Centre for the Study of Gambling
University of Salford
Greater Manchester, M5 4WT, United Kingdom
RANDOMNESS OF
EUROMILLIONS DRAWS
A Report for the
National Lottery Commission
January 2010Randomness of EuroMillions Draws:
Executive Summary
The National Lottery Commission invited the Centre for the Study of
Gambling (University of Salford) to establish whether there are any elements of
non-randomness within EuroMillions. Concentrating on the first four years of
EuroMillions draws, from 13th February 2004 to 8th February 2008 inclusive,
specific objectives were to test whether:
(a) there is equality of frequency for each EuroMillions number drawn;
(b) each EuroMillions draw is independent of preceding draws;
(c) there is any bias for or against tickets sold in particular countries;
(d) the frequency of top tier winners is what would be expected from
random draws.
Players in any particular draw select five main numbers from 1 to 50 and
two “Lucky Star” numbers from 1 to 9. They may win various prizes by matching
their chosen numbers with selected combinations drawn. 50% of ticket sales are
allocated to the Common Prize Fund, which is distributed among different prize
tiers according to pre-specified percentages. There are no fixed prize tiers. Our
investigations began with the results of an exploratory data analysis. The findings
here were that there appear to be no obvious discrepancies and that all entries in
the database appear to be correct.
Perhaps the most important hypothesis tests are whether the frequencies of
the five main numbers drawn agree with the null hypothesis that the five main
balls are drawn at random and whether the frequencies of the Lucky Star numbers
drawn agree with the null hypothesis that the Lucky Star balls are drawn at
2random. Neither of our modified chi-square goodness-of-fit tests was significant
at the 5% level, providing evidence in support of random EuroMillions drawings.
We next conducted a set of tests for the sequential independence of
EuroMillions draws. Based on the gaps between successive selections of the
specific main numbers 1 to 50, we employed chi-square goodness-of-fit tests to
compare observed and expected frequencies of possible gap sizes. Using the
Bonferroni method for multiple comparisons to adjust the critical p-value and
allow for multiplicity, the overall test was not significant at the 5% level and we
conclude that successive EuroMillions main number draws appear to be
independent of one another.
We performed a similar analysis for the nine Lucky Star numbers and this
overall test was not significant at the 5% level, so we conclude that successive
EuroMillions Lucky Star number draws appear to be independent of one another.
As a further check for the possibility of serial dependence, we performed runs
tests for low valued balls and neither of the runs tests for main numbers and
Lucky Stars was significant. This again provides evidence that the outcomes of
specific draws do not depend on preceding draw results.
Next, we generated graphical comparisons of observed and expected
frequency distributions for the order statistics of the five main numbers and for
the order statistics of the two Lucky Stars. The observed frequencies resemble the
expected frequencies in each case, further supporting our formal tests of
randomness. Then we performed several tests for randomness of the five main
numbers and two Lucky Stars, based upon the means and standard deviations of
the sums of numbers in each draw, and upon the parities of the numbers in each
draw. None of these tests was significant at the 5% level, providing further
evidence in support of random drawings.
3We then investigated whether the first four years of EuroMillions draw
history provide any evidence of bias for or against tickets sold in particular
countries. By calculating sample correlation coefficients between the sales
figures for each country and (i) the numbers of jackpot winners, (ii) the total
numbers of winners and (iii) the total prize monies, we were able to measure the
associations between these pairs of variables. As illustrated by scatter plots and
regression fits, these correlations are all close to one with no clear outliers,
providing evidence to support the hypothesis of no bias.
The final stage was to consider whether the frequency of top tier winners
(match five main numbers plus two Lucky Stars) agrees with expectation under
the assumption that all EuroMillions procedures operate randomly. Knowledge of
the coverage for each draw provided sufficient information to analyse the
frequency of draws with no top tier winners, separately for the U.K. game only
and for the aggregate game across all participating countries. Both chi-square
goodness-of-fit tests showed that there were no significant differences between
the observed and expected frequencies. These results provide supporting
evidence that the process of generating winning numbers is independent of the
pattern of numbers chosen by the playing population.
Although we followed the convention of testing for significance at the 5%
level throughout this investigation, it is worth noting that none of the results we
obtained would be significant even at the 10% level. From all of these results, it
is clear that the first four years of draw history provide no evidence that there are
any elements of non-randomness within EuroMillions.
4Randomness of EuroMillions Draws:
Detailed Findings
ABSTRACT
The researchers conducted a variety of hypothesis tests for randomness of the
first four years of EuroMillions draws up to and including 8th February 2008. These
cover aspects relating to the frequencies of numbers drawn, possible dependencies
between draws, comparisons across participating countries and frequencies of top
tier winners. Overall, our analysis supports the hypothesis of randomness. That is,
there is no statistical evidence of non-randomness in the EuroMillions draws.
1. Introduction
The U.K. National Lottery Commission (N.L.C.) invited the Centre for the
Study of Gambling (University of Salford) to establish whether there are any
elements of non-randomness within EuroMillions. The EuroMillions draws
generally take place in Paris on Friday evenings and there have been 209 draws
since the inception of the game on 13th February 2004 up to and including 8th
February 2008. The N.L.C. provided the researchers with data relating to these
first four years of operation and specific objectives were to test whether:
(a) there is equality of frequency for each EuroMillions number drawn;
(b) each EuroMillions draw is independent of preceding draws;
(c) there is any bias for or against tickets sold in particular countries;
(d) the frequency of top tier winners is what would be expected from
random draws.
5We begin with an exploratory analysis of the data in order to check that
there are no unusual, suspicious or incorrect observations. Then we perform a
range of suitable hypothesis tests for randomness to answer the above questions.
Some of these procedures are modifications of standard statistical methods
described in textbooks such as Miller and Miller (2004) and Rice (2007). Others
derive from specific recommendations made by Joe (1993) and Haigh (1997), and
from the Royal Statistical Society’s early reports into the randomness of the
lottery (2000, 2002).
For consistency, we follow the general methodology adopted for previous
National Lottery Commission reports on the randomness of Lotto Draws (2004),
Lotto Lucky Dip (2005), Thunderball Draws (2005), Lotto Extra Draws (2005)
and EuroMillions Lucky Dip (2010) produced by the Centre for the Study of
Gambling (University of Salford). The sample sizes available to us are sufficient
to enable standard, powerful tests of all the above hypotheses and so to produce
statistically valid and reliable conclusions. We choose to work with the
conventional 5% significance level, corresponding with a 95% confidence level,
thereby maintaining consistency with most published statistical analyses and
ensuring ease of interpretation.
The tests for randomness that we use in this report complement each other
and cover the most important features relating to the randomness of EuroMillions
draws. Of primary importance, we use a standard chi-square goodness-of-fit test,
modified to allow for sampling without replacement, to assess whether the
winning balls are equally likely to be chosen and whether any observed variability
is acceptable under the assumption of random sampling.
To monitor for any serial dependence among the draws, we use standard
chi-square goodness-of-fit tests based on the numbers of draws (gaps) between
successive appearances of each number, separately for the main draw numbers
and the Lucky Stars. These tests are capable of identifying any irregularities
6arising due to time trends. As a separate test is needed for each number, we apply
multiplicity adjustments to the test results in order to avoid false alarms.
We then conduct a variety of other analyses and tests that are capable of
detecting other patterns of non-randomness. An infinite number of possibilities
exist and we select several convenient and complementary techniques that permit
a good overall assessment of the data. Based on the results of each draw, these
include monitoring and testing: the ordered ball values; the sum of the ball values;
the odd/even spread of ball values.
We also check for any bias across the different participating countries. We
do this using sample correlation coefficients and fitted regression lines for the
numbers of jackpot winners, numbers of other winners and values of prizes.
Finally, we use a standard binomial test to assess whether the frequencies of
jackpot winners in the U.K. and overall agree with what would be expected under
random drawings, using knowledge of the coverage rates for each draw.
2. Exploratory Data Analysis
The EuroMillions game involves nine European countries: Austria;
Belgium; France; Ireland; Luxembourg; Portugal; Spain; Switzerland; U.K. (and
the Isle of Man). In the U.K., a customer pays an amount in pounds sterling (to
date £1.50) that reasonably approximates the equivalent of two Euros, in order to
enter the EuroMillions game. Any excess payments (or any deficiencies)
associated with this not being an exact conversion at the exchange rate prevailing
on the day of the draw is returned to (or collected from) U.K. players via an
adjustment to amounts paid out as lower tier prizes in the U.K.
7Winning Approximate Prize Tier
Combination Probability Allocations
Match 5 and 2 Lucky Stars 1 / 76,275,360 32.0%
Match 5 and 1 Lucky Star 1 / 5,448,240 7.4%
Match 5 1 / 3,632,160 2.1%
Match 4 and 2 Lucky Stars 1 / 339,002 1.5%
Match 4 and 1 Lucky Star 1 / 24,214 1.0%
Match 4 1 / 16,143 0.7%
Match 3 and 2 Lucky Stars 1 / 7,705 1.0%
Match 3 and 1 Lucky Star 1 / 550 5.1%
Match 2 and 2 Lucky Stars 1 / 538 4.4%
Match 3 1 / 367 4.7%
Match 1 and 2 Lucky Stars 1 / 102 10.1%
Match 2 and 1 Lucky Star 1 / 38 24.0%
Aggregate 1 / 24 94%
Table 1: winning combinations, probabilities and prize allocations.
Players in any particular draw select five main numbers from 1 to 50 and
two “Lucky Star” numbers from 1 to 9. They may win various prizes by matching
their chosen numbers with selected combinations drawn, as shown in Table 1.
50% of ticket sales are allocated to the Common Prize Fund. 6% of the Common
Prize Fund is allocated to the Reserve Fund to supplement the jackpot pool. The
remainder (94%) of the Common Prize Fund is allocated to different prize tiers
according to the percentages set out in Table 1. There are no fixed prize tiers.
The equation for calculating the probability of a “Match x and y Lucky
Stars” combination is
8 45 5 7 2
5 − x x 2 − y y
p( x, y ) = × ; x = 0,1, K ,5 ∩ y = 0,1,2 (1)
50 9
5
2
where
n n!
= (2)
m m!(n − m )!
is the number of combinations of m items that can be selected from a total of n
items. Any player who matches all five main numbers and both Lucky Star
numbers wins a share of the top-tier jackpot prize.
Table 2 and Figure 1 present a tally chart and bar chart illustrating the
observed frequencies of occurrence f i for each of the numbers i of the five main
balls selected during the first 209 draws up to and including 8th February 2008. A
cursory glance at these data is sufficient to show that there are no obviously false
records in this part of the database: only the natural numbers from one to fifty are
recorded, none of the observed frequencies are unreasonable values and the
observed frequencies sum to 5 × 209 = 1,045 . The expected frequency for each of
the numbers is 1,045 ÷ 50 = 20 ⋅ 9 and the observed frequencies in the bar chart
display a good scatter about this value. Although only nine of the first 209 draws
selected ball 46, we shall later verify that this is in accordance with natural
random variation.
9i fi i fi i fi i fi i fi
1 28 11 24 21 25 31 17 41 24
2 18 12 27 22 17 32 19 42 19
3 28 13 16 23 25 33 18 43 21
4 21 14 23 24 17 34 21 44 24
5 19 15 24 25 22 35 21 45 21
6 23 16 20 26 22 36 24 46 9
7 21 17 16 27 17 37 26 47 21
8 24 18 17 28 14 38 21 48 18
9 22 19 24 29 20 39 15 49 23
10 23 20 16 30 20 40 20 50 30
Table 2: observed frequencies of the five main numbers drawn.
30
25
20
Frequency
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
Number
Figure 1: observed frequencies of the five main numbers drawn.
10Table 3 and Figure 2 present a tally chart and bar chart illustrating the
observed frequencies of occurrence f i for each of the numbers i of the two
Lucky Star balls selected during the first 209 draws up to and including 8th
February 2008. These frequencies sum to 2 × 209 = 418 and a cursory glance at
these data is sufficient to show that there are no obviously false records in this
part of the database: only the natural numbers from one to nine are recorded and
none of the observed frequencies are unreasonable values. The expected
frequency for each of the numbers is 418 ÷ 9 = 46 ⋅ 4& and the observed
frequencies in the bar chart display a good scatter about this value.
i 1 2 3 4 5 6 7 8 9
fi 53 40 50 37 49 52 47 47 43
Table 3: observed frequencies of the two Lucky Star numbers drawn.
50
40
Frequency
30
20
10
0
0 1 2 3 4 5 6 7 8 9
Number
Figure 2: observed frequencies of the two Lucky Star numbers drawn.
The histogram in Figure 3 displays the sales per draw in millions of pounds
for the U.K. EuroMillions game. Clearly, a large majority of draws attracted sales
of below ten million pounds, though the sales increased to a peak of over fifty
million pounds when there were consecutive rollovers and event draws on offer.
11This is also evident from the subsequent scatter plot in Figure 4, where some
outliers are evident.
100
80
Number of Draws
60
40
20
0
0 5 10 15 20 25 30 35 40 45 50 55
U.K. Sales (£m)
Figure 3: sales per draw for the U.K. EuroMillions game.
50
40
U.K. Sales (£m)
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11
Number of Consecutive Rollovers
Figure 4: the relationship between U.K. sales and consecutive rollovers.
12The most prominent of these are two particularly large U.K. sales amounts
for first time draws (no rollovers) of about £34m on 28th September 2007 and
about £17m on 9th February 2007. On further inspection, these both correspond to
event draws, with jackpot pools of 130m Euros and 100m Euros respectively
guaranteed by the Reserve Fund. An even more outlying U.K. sales amount of
about £49m occurred on 8th February 2008 after one rollover. This corresponds to
an event draw with a guaranteed jackpot pool of 130m Euros, on a draw that
would otherwise have represented a single rollover. Apart from these instances,
there are no particularly unusual observations that require closer inspection.
Three graphs follow to display the coverage rates and their relationships to
sales. Coverage refers to the proportion of all possible combinations that at least
one player selects in any particular draw.
60
50
Number of Draws
40
30
20
10
0
0 10 20 30 40 50 60 70 80 90 100
Total Coverage (%)
Figure 5: total coverage across all EuroMillions countries.
Figure 5 is a histogram, which displays the aggregate coverage rates across
all participating countries for the first four years of draws, expressed as
percentages. This is useful for assessing whether the frequency of rollovers is
13consistent with the hypothesis of random drawings. Conscious selection is likely
to be present, whereby many players do not select numbers and combinations
randomly. However, this does not affect the randomness of the draws and the
Lucky Dip facility helps to spread the players’ choices across a large proportion
of the possible combinations.
40
30
U.K. Coverage (%)
20
10
0
0 10 20 30 40 50 60
U.K. Sales (£m)
Figure 6: relationship between coverage and sales for U.K. EuroMillions.
The expected coverage for any draw correlates positively with the number
of tickets sold for that draw. Figure 6 presents a scatter plot of U.K. coverage
against U.K. sales. The relationship between the variables is strikingly regular:
near linear with a hint of concavity. The close approximation to linearity arises
because the number of tickets sold in the U.K. is well below the number of
possible combinations.
1460
Sales Coverage
50
U.K. Coverage (%)
40
U.K. Sales (£m)
30
20
10
0
1 21 41 61 81 101 121 141 161 181 201
Draw Num ber
Figure 7: time trends of sales and coverage for U.K. EuroMillions.
Finally, Figure 7 presents two time series plots, which illustrate how the
U.K. sales (£m) and U.K. coverage (%) variables have progressed over time. The
obvious peaks coincide with consecutive rollovers and event draws. None of
these graphs identifies any unusual observations that cast doubt on the accuracy of
the recorded data. It is interesting to note that U.K. sales were relatively low for
the first year, as only three countries participated during for the first thirty-four
weeks and prize values were correspondingly less.
We conclude this section with another time series plot, in Figure 8, which
displays the numbers of Match 5 plus 2 Lucky Stars jackpot winners over the first
209 draws. During these first four years, there were 89 jackpot winners across all
nine countries, including 10 from the U.K. Total sales during this period
amounted to about 15,066 million Euros, compared with the U.K. sales of about
1,898 million Euros.
155
U.K. Total
4
Number of Jackpot Winners
3
2
1
0
1 21 41 61 81 101 121 141 161 181 201
Draw Num ber
Figure 8: time series plots of jackpot winners.
This is a sales ratio of about 8:1, which roughly equates with the ratio of
about 9:1 for jackpot winners and so indicates that the U.K. is rewarded in accord
with mathematical expectation. We present a formal statistical analysis of the
frequency of jackpot winners later in this report. However, the time series plot in
Figure 8 appears to be compatible with what one might expect from chance alone.
Note that there were no draws with multiple jackpot winners early on, when fewer
countries participated.
3. Testing whether there is Equality of Frequency
for each EuroMillions Number Drawn
Perhaps the most important tests for the EuroMillions game are whether the
observed frequencies of the fifty main numbers and nine Lucky Star numbers
accord with what would be expected from random drawings.
16We test the equality of marginal frequencies for the five main numbers
chosen in EuroMillions draws using a chi-square goodness-of-fit test modified to
allow for sampling without replacement. Defining m = 5 , M = 50 and D = 209 ,
the test statistic is
M 2 m2D2
(M − 1)M ∑ f i −
i =1 M
T= (3)
(M − m)Dm
where ball i is selected f i times for i = 1,2,K , M . This is equivalent to
(M −1) ÷ (M − m) multiplied by the usual chi-square goodness-of-fit statistic; the
expected frequencies in this case are all m × D ÷ M = 20 ⋅ 9 . We compare this test
statistic with the critical value for the rejection region in the upper tail of the
χ 2 (M − 1) distribution.
The null hypothesis is that the observed frequencies accord with random
drawings and the alternative hypothesis is that they do not. For the first 209
draws, we have T ≈ 40 ⋅ 7 on 49 degrees of freedom, corresponding to a p-value
of p ≈ 0 ⋅ 796 . As this is greater than 0 ⋅ 05 , the test is not significant and we do
not reject the null hypothesis at the 5% level of significance. We conclude that
there is no evidence to suggest the possibility of bias in the selection of the five
main numbers in the EuroMillions draws.
It is important to realize that the EuroMillions draw operators use a variety
of machines and ball sets to avoid systematic bias. Furthermore, p ≈ 0 ⋅ 796
means that when no bias is present, the probability of observing a frequency
distribution at least as non-uniform as the one actually observed is approximately
four fifths. Even though only nine of the first 209 draws selected ball 46, this
modified chi-square goodness-of-fit test clearly demonstrates that this is in
accordance with natural random variation.
17Next, we consider the Lucky Star numbers chosen in EuroMillions draws
and test the null hypothesis that the draw procedures are equally likely to select
each of the nine numbers as Lucky Stars. If all balls are equally likely for
selection as Lucky Stars, the expected frequency is 2 × 209 ÷ 9 = 46 ⋅ 4& for each of
the nine possible numbers. We again compare the observed frequencies with
these expected frequencies by means of a modified chi-square goodness-of-fit
test, using Equation (3) after re-defining m = 2 and M = 9 .
The observed test statistic is T ≈ 5 ⋅ 81 on 8 degrees of freedom,
corresponding to a p-value of p ≈ 0 ⋅ 668 . Thus when no bias is present, the
probability of observing a frequency distribution at least as non-uniform as the
one actually observed is more than two thirds. Consequently, the test is not
significant at the 5% level and we conclude that there is no evidence to suggest
that the selection of the Lucky Star numbers is anything but random.
4. Testing whether each EuroMillions Draw is
Independent of Preceding Draws
For any fixed number i = 1,2, K , M let g 1 denote the number of draws until
ball i first appears. Similarly, let g 2 , g 3 , K be the numbers of draws between
later successive appearances of ball i . Under the null hypothesis that there is
independence between the draws, these gaps are independent geometric random
variables, with probability mass function
g −1
m m
p ( g ) = 1 − ; g = 1,2,3, K . (4)
M M
18This result enables us to perform a standard chi-square goodness-of-fit test of the
null hypothesis for each fixed number.
This involves comparing the observed gap frequencies with those expected
under the null hypothesis using Equation (4), by evaluating the test statistic
T =∑
(obs. − exp.)2 (5)
exp.
where summation is over the number of gap categories considered. For each of
the M possible numbers, we calculate the expected frequencies for the chosen
categories by scaling the corresponding probabilities from Equation (4) by the
total number of observed gaps for that number, ignoring the final censored gap
interval unless it clearly belongs to the uppermost category with an open interval.
Under the null hypothesis of independent draws, the test statistic in Equation (5)
asymptotically has a chi-square distribution with degrees of freedom equal to the
number of categories minus one. The test is one sided and we reject the null
hypothesis if the test statistic lies in the upper tail of this distribution.
Based on the draw history available at this time and to avoid unnecessary
complexity, we define only two categories and do so by referring to the median
gap length assuming independent draws. The resulting categories correspond to
gap lengths in the ranges g = 1,2, K , G and g = G + 1, G + 2, G + 3, K where we
select the value of G to minimise the absolute difference metric
G
1
d= ∑ p(g ) − 2
g =1
(6)
from Equation (4). This ensures that the two categories are approximately equally
likely, which maximises the power of this test.
19Furthermore, as there are only two categories, we replace the approximate
chi-square goodness-of-fit tests in Equation (5) by exact binomial tests for
improved accuracy, based on two-tailed probabilities from the binomial
distribution with probability mass function
t
p( y ) = q y (1 − q ) ;
t−y
y = 0,1,K, t (7)
y
where y is the observed number of gaps in category 1, t is the total number of
gaps observed in categories 1 and 2, and
G
q = ∑ p( g ) (8)
g =1
is the cumulative probability of gap lengths based on Equation (4).
These tests are dependent for i = 1,2, K , M and there is clear multiplicity.
The dependence means that these are only approximate tests of the null
hypothesis, though they are still relevant and informative. The multiplicity
problem is easily resolved by applying multiple comparisons procedures. We
choose the Bonferroni adjustment for this purpose because of its simple
interpretation, though other less-conservative procedures are available. To apply
this adjustment, a test for independence of draws at the 5% level of significance
involves comparing each of the M p-values with 0 ⋅ 05 ÷ M rather than with the
unadjusted value of 0 ⋅ 05 .
For the m = 5 main number balls selected in any EuroMillions draw, we set
M = 50 as before, in which case the median gap length satisfies G = 7 by using
Equation (4) to minimise the metric in Equation (6). Consequently, we define our
two categories as g = 1,2, K ,7 and g = 8,9,10, K . We present the results of our
20binomial gap tests for serial dependence of the main numbers drawn in Table 4,
which gives details of the observed frequencies y i in category 1, the total
observed frequencies t i in categories 1 and 2, and the p-value p i for each fixed
number i .
i y i (t i ) pi i y i (t i ) pi i y i (t i ) pi i y i (t i ) pi
1 20(29) 0.102 14 13(23) 0.837 27 7(18) 0.373 40 6(20) 0.077
2 7(18) 0.373 15 17(25) 0.164 28 7(15) 0.864 41 16(25) 0.325
3 21(29) 0.043 16 7(20) 0.189 29 7(21) 0.130 42 11(20) 0.979
4 9(22) 0.399 17 8(16) 1.000 30 10(20) 1.000 43 10(22) 0.676
5 9(20) 0.675 18 6(18) 0.172 31 8(17) 0.856 44 15(24) 0.420
6 10(23) 0.531 19 15(24) 0.420 32 9(20) 0.675 45 12(21) 0.815
7 14(22) 0.389 20 9(16) 0.942 33 9(19) 0.848 46 0(9) 0.003
8 12(24) 0.991 21 15(25) 0.562 34 11(23) 0.833 47 12(21) 0.815
9 10(22) 0.676 22 9(17) 1.000 35 10(22) 0.676 48 7(19) 0.268
10 14(23) 0.533 23 13(25) 1.000 36 16(24) 0.222 49 14(23) 0.533
11 14(25) 0.857 24 10(17) 0.763 37 13(27) 0.820 50 19(30) 0.298
12 15(27) 0.876 25 14(22) 0.389 38 10(21) 0.840
13 6(17) 0.250 26 12(23) 1.000 39 6(16) 0.355
Table 4: observed frequencies and p-values of gap tests for main numbers.
To interpret the results following a Bonferroni adjustment for multiple
comparisons, we compare the p-values with the adjusted significance level
0 ⋅ 05 ÷ 50 = 0 ⋅ 001 and reject the null hypothesis at the 5% level if any of the p-
values is less than this value. From Table 4, we see that none of the p-values is
less than 0 ⋅ 001 , so the test is not significant at the 5% level and we conclude that
there is no evidence of serial dependence among the main numbers drawn in the
EuroMillions game.
21For the m = 2 Lucky Star number balls selected in any EuroMillions draw,
we set M = 9 , in which case the median gap length satisfies G = 3 by using
Equation (4) to minimise the metric in Equation (6). Consequently, we define our
two categories as g = 1,2,3 and g = 4,5,6, K . We present the results of our
binomial gap tests for serial dependence of the Lucky Star numbers drawn in
Table 5, which gives details of the observed frequencies y i in category 1, the total
observed frequencies t i in categories 1 and 2, and the p-value p i for each fixed
number i .
i 1 2 3 4 5 6 7 8 9
y i (t i ) 26(53) 20(41) 32(51) 21(37) 24(50) 30(53) 24(47) 28(47) 22(43)
pi 0.666 0.704 0.206 0.767 0.575 0.695 0.908 0.446 0.932
Table 5: observed frequencies and p-values of gap tests for Lucky Star numbers.
To interpret the results following a Bonferroni adjustment for multiple
comparisons, we now compare the p-values with the adjusted significance level
0 ⋅ 05 ÷ 9 ≈ 0 ⋅ 006 and reject the null hypothesis at the 5% level if any of the p-
values is less than this value. From Table 5, we see that none of the p-values is
less than 0 ⋅ 006 , so the test is not significant at the 5% level and we conclude that
there is no evidence of serial dependence among the Lucky Star numbers drawn in
the EuroMillions game.
We conclude this section with two further tests for serial dependence of
EuroMillions draws, based on runs of small and large numbers. One of these is
for the main numbers drawn and the other is for the Lucky Stars numbers drawn.
Consider first the main numbers. For each draw, we count the number of
low valued balls, which we define as 1 to 25. Then we classify each draw as low
or high valued, depending on whether or not it comprises at least three low valued
22balls. Now define L and H to be the total frequencies of low and high valued
draws, with L + H = D where D = 209 as before. Then, the total number of runs
(successions of identical classifications) R has the asymptotic normal distribution
2 LH 2 LH (2 LH − D )
R ~& N + 1, (9)
D D 2 (D − 1)
and so we can perform a standard normal hypothesis test for trends based on the
test statistic
R − µR
z= (10)
σR
where µ R and σ R are the mean and standard deviation of R from Relation (9).
For the available draw history, we observe L = 111 , H = 98 and R = 103 ,
corresponding to the test statistic z ≈ 0 ⋅ 292 from Equation (10) and a p-value of
p ≈ 0 ⋅ 770 . As the latter exceeds 0 ⋅ 05 , the test is not significant at the 5% level
and so this test provides no evidence against serial independence of the main
numbers drawn.
Now consider the Lucky Stars. For each draw, we count the number of low
valued balls, which we define as 1 to 5. Then we classify each draw as low or
high valued, depending on whether or not it comprises two low valued balls.
Based on the available draw history, we observe L = 58 , H = 151 and R = 82 ,
corresponding to the test statistic z ≈ −0 ⋅ 486 from Equation (10) and a p-value of
p ≈ 0 ⋅ 627 . As the latter exceeds 0 ⋅ 05 , the test is not significant at the 5% level
and so this test provides no evidence against serial independence of the Lucky
Star numbers drawn.
235. Other Tests for Randomness of the Five Main
Numbers and the Lucky Stars Drawn
We now present several further tests for randomness of the five main
EuroMillions numbers and the two Lucky Star numbers drawn. We first conduct
five complementary analyses of the EuroMillions main number combinations
generated, based upon the theoretical probability distributions of the order
statistics within each draw under the hypothesis of randomness. That is, we
derive the actual distributions for the smallest (first order statistic), the next
smallest (second order statistic) and so on, of the five main numbers generated.
Then we compare the histograms based on actual observed order statistics with
the corresponding hypothetical distributions. Whereas the tests in Section 3 are
concerned with frequencies of the individual numbers drawn, the tests in this
section are able to detect patterns and clustering within the combinations drawn.
In D = 209 random draws of five numbers from 1 to 50, the frequency of
appearances X by any particular number has probability mass function
D
p (x ) = p x (1 − p ) ; x = 0,1, K , D
D− x
(11)
x
where p = 5 ÷ 50 = 0 ⋅ 1 . Although this binomial distribution forms the basis of
the test constructed in Section 3, it is not suitable for the order statistics, so we
derive the corresponding hypothetical distributions from first principles.
The first order statistic X (1) in a combination of five main EuroMillions
numbers is the smallest of those five numbers. The second order statistic X (2 ) is
the next smallest of those five numbers and so on. By considering elementary
24combinatorics, we find that a general formula for the probability mass function of
the k th order statistic takes the form
x − 1 50 − x
k − 1 5 − k
P (X ( k ) = x) = ; x = k , k + 1, K , k + 45 (12)
50
5
for each of the values k = 1,2, K ,5 .
Based upon this probability distribution, we can determine the expected
frequencies for all five order statistics under the null hypothesis that the
EuroMillions main number combinations are random. We then compare our
observed frequencies with these by means of the graphs in Figures 9 to 13, which
display the observed frequencies as markers and the expected frequencies as bars.
Although we could perform formal chi-square goodness-of-fit tests to assess
whether the order statistics accord with the assumption of random drawings, the
graphs clearly illustrate that there are no consistent patterns that might indicate
unusual behaviour and so we do not proceed with a formal analysis of this aspect.
30
25
number of draws
20
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
first order statistic
Figure 9: obs. and exp. frequencies for first order statistic of main numbers.
2530
25
number of draws 20
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
second order statistic
Figure 10: obs. and exp. frequencies for second order statistic of main numbers.
30
25
number of draws
20
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
third order statistic
Figure 11: obs. and exp. frequencies for third order statistic of main numbers.
30
25
number of draws
20
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
fourth order statistic
Figure 12: obs. and exp. frequencies for fourth order statistic of main numbers.
2630
25
number of draws 20
15
10
5
0
0 5 10 15 20 25 30 35 40 45 50
fifth order statistic
Figure 13: obs. and exp. frequencies for fifth order statistic of main numbers.
We now repeat this analysis for the EuroMillions Lucky Stars. The
probability mass function of the k th order statistic in a combination of two Lucky
Star numbers drawn takes the form
x − 1 9 − x
k − 1 2 − k
P (X ( k ) = x) = ; x = k , k + 1, K , k + 7 (13)
9
2
for k = 1,2 . Based upon this probability distribution, we can determine the
expected frequencies for both order statistics under the null hypothesis that the
EuroMillions Lucky Star number combinations are random.
We then compare our observed frequencies with these by means of the
graphs in Figures 14 and 15, which display the observed frequencies as markers
and the expected frequencies as bars. Although we could perform formal chi-
square goodness-of-fit tests to assess whether the order statistics accord with the
assumption of random drawings, the graphs clearly illustrate that there are no
consistent patterns that might indicate unusual behaviour and so we do not
proceed with a formal analysis of this aspect.
2750
40
number of draws
30
20
10
0
0 1 2 3 4 5 6 7 8 9
first order statistic
Figure 14: obs. and exp. frequencies for first order statistic of Lucky Stars.
50
40
number of draws
30
20
10
0
0 1 2 3 4 5 6 7 8 9
second order statistic
Figure 15: obs. and exp. frequencies for second order statistic of Lucky Stars.
We now consider several tests based on the sum of the EuroMillions
numbers drawn. Firstly, define n ij to be the main number associated with ball i
in draw j and the sum of the numbers selected in draw j by
m
s j = ∑ nij . (14)
i =1
Under the null hypothesis of random draws, the mean of s j is
28m(M + 1)
µ= (15)
2
and the variance of s j is
m(M + 1)(M − m )
σ2 = . (16)
12
We now use the central limit theorem to derive asymptotic sampling distributions
for the sample mean U and sample variance V of s j for j = 1,2, K , D . These
sampling distributions provide two two-sided tests of the null hypothesis that the
EuroMillions numbers are drawn at random:
U −µ
~& N (0,1) ; (17)
σ D
(D − 1)V ~& χ 2 (D − 1) . (18)
σ 2
With D = 209 , M = 50 and m = 5 for the main numbers drawn, we have
µ = 127 ⋅ 5 and σ 2 = 956 ⋅ 25 , and observed sample statistics U ≈ 125 and
V ≈ 944 , corresponding to p-values of p ≈ 0 ⋅ 300 and p ≈ 0 ⋅ 923 from Relations
(17) and (18) respectively. Consequently, neither of these tests is significant at
the 5% level, again providing evidence in favour of randomness of the main
numbers drawn in the EuroMillions game.
Similarly, with D = 209 , M = 9 and m = 2 for the Lucky Star numbers
drawn, we have µ = 10 and σ 2 = 11 ⋅ 6& , and observed sample statistics U ≈ 9 ⋅ 95
and V ≈ 11⋅ 9 , corresponding to p-values of p ≈ 0 ⋅ 840 and p ≈ 0 ⋅ 813 from
Relations (17) and (18) respectively. Consequently, neither of these tests is
significant at the 5% level, again providing evidence in favour of randomness of
the Lucky Star numbers drawn in the EuroMillions game.
29We now consider two tests based upon the observed odd and even
combinations that occur in given EuroMillions draws. For the first of these,
define e j to be the number of even numbers among the m = 5 main numbers
selected in draw j based on M = 50 possible numbers. Under the assumption of
random draws, the sampling distribution for e j is hypergeometric with probability
mass function
r M − r
e m − e
p(e ) = ; e = 0,1, K , m (19)
M
m
where r = 25 , the number of even numbers between 1 and 50 inclusive.
Consequently, we can perform a chi-square goodness-of-fit test using the
test statistic
T =∑
(obs. − exp.)2 (20)
exp.
to see whether our observed frequencies of even numbers per draw agree with
what is expected by chance alone. This time, there are m = 5 degrees of freedom
for the test. Based on the first 209 draws, our observed test statistic is T ≈ 3⋅ 16
corresponding to a p-value of p ≈ 0 ⋅ 675 . As this value exceeds 0 ⋅ 05 , the result
is not significant at the 5% level and this test again supports the null hypothesis of
randomness of the main numbers in EuroMillions draws.
Repeating this test for the Lucky Stars numbers, we set M = 9 and m = 2
so r = 4 and we refer to the χ 2 (2 ) sampling distribution. Our observed test
statistic is now T ≈ 1⋅ 09 corresponding to a p-value of p ≈ 0 ⋅ 580 . As this value
exceeds 0 ⋅ 05 , the result is not significant at the 5% level and this test again
30supports the null hypothesis of randomness of the Lucky Star numbers in
EuroMillions draws.
6. Testing whether there is any Bias for or against
Tickets Sold in Particular Countries
In Table 6, we present details of the total EuroMillions sales over the first
209 draws in millions of Euros, divided up into totals for each of the nine
participating countries. In order to assess whether inhabitants of each country can
expect a proportion of winning tickets similar to the proportion of total sales they
contribute, we present details of the numbers of jackpot winners, total numbers of
winners in millions and total prize monies awarded in millions of Euros.
Total number Total value
Total sales Number of of winners of prizes
Country (mEuros) jackpot winners (millions) (mEuros)
U.K. 1,898 10 40 987
France 3,966 27 84 2,067
Spain 3,141 19 66 1,617
Belgium 941 4 20 563
Ireland 344 1 7 257
Portugal 3,302 21 69 1,620
Luxembourg 96 0 2 27
Austria 520 2 11 226
Switzerland 857 5 18 476
Total 15,066 89 318 7,841
Table 6: breakdown of EuroMillions sales, winners and prizes across countries.
Table 7 presents the same information expressed as percentages of the totals
across all participating countries, rather than as absolute values. This enables
31easier comparisons to assess whether there is any bias for or against particular
countries. These percentages are consistent for all countries, providing no
evidence of any preferential bias.
Number of Total number Total value
Total sales jackpot winners of winners of prizes
Country (%) (%) (%) (%)
U.K. 13 11 13 13
France 26 30 26 26
Spain 21 21 21 21
Belgium 6 4 6 7
Ireland 2 1 2 3
Portugal 22 24 22 21
Luxembourg 1 0 1 0
Austria 3 2 3 3
Switzerland 6 6 6 6
Total 100 100 100 100
Table 7: percentages of EuroMillions sales, winners and prizes across countries.
Our statistical analysis involves calculating the sample correlation
coefficients between each of the three outcome measures and the sales figures by
country in Table 6. Correlations must lie in the closed interval [− 1, 1] and, if the
outcomes of EuroMillions draws treat tickets randomly (irrespective of where
they were bought), these correlations should all be close to unity. In the order
presented above, these correlations are 0 ⋅ 996 , 1⋅ 000 and 0 ⋅ 998 respectively,
which clearly support the claims of no bias.
Figures 16, 17 and 18 display these close relationships graphically, together
with superimposed regression lines. All regression fits are highly significant,
which we would expect if all participating countries are treated randomly.
Consequently, we conclude that the first four years of EuroMillions draw history
32provide no evidence of any bias for or against tickets sold in particular countries.
Specifically, the EuroMillions game procedures appear to treat the U.K. on an
equal basis as any other participating country.
25
jackpot winners
20
15
U.K.
10
5
0
0 1000 2000 3000 4000
sales (mEuros)
Figure 16: scatter plot of jackpot winners against sales across countries.
80
total winners (millions)
60
U.K.
40
20
0
0 1000 2000 3000 4000
sales (mEuros)
Figure 17: scatter plot of total winners against sales across countries.
332000
total prizes (mEuros)
1500
U.K.
1000
500
0
0 1000 2000 3000 4000
sales (mEuros)
Figure 18: scatter plot of total prizes against sales across countries.
7. Testing whether the Frequency of Top Tier
Winners is as Expected
In a draw with N random entries, the number of top tier winners w has a
binomial distribution with probability mass function
N
p(w) = p w (1 − p )
N −w
; w = 0,1, K , N (21)
w
where p = 1 ÷ 76,275,360 from Section 2. From this, the probability that a given
draw has no top tier winners is p (0 ) = (1 − p ) .
N
For illustration, the mean sales per draw of U.K. EuroMillions over the first
four years is about £6,809,653 , which equates to about N = 4,539,769 entries per
draw. This suggests that the probability of no U.K. top tier winners should be
roughly 0 ⋅ 940 , equivalent to about 94% of draws on average, except that some
players might possibly select the same number combinations. Over the course of
the first 209 draws, there was at least one U.K. jackpot winner on eight occasions.
34This corresponds to about 96% of draws having no U.K. top tier winners, which
would be perfectly acceptable under the assumption that the spatial distribution of
jackpot winners is random.
Similarly, the mean aggregate sales per draw of EuroMillions over the first
four years is about 72,084,098 Euros, which equates to about N = 36,042,049
entries per draw. This suggests that the probability of no top tier winners from
any country should be roughly 0 ⋅ 527 , equivalent to about 53% of draws on
average, except that some players would likely select the same number
combinations. Over the course of the first 209 draws, there was at least one
aggregate jackpot winner on sixty-four occasions. This corresponds to about 69%
of draws having no aggregate top tier winners, which would be acceptable if the
distribution of jackpot winners were random.
160
140
number of draws
120
100
80
60
40
20
0
-1 0 1 2 3 4 5 6
number of top tier winners
Figure 19: observed and expected frequencies of aggregate jackpot winners.
However, it is well known that most players do not in fact select their
lottery numbers at random and this “conscious selection” has the effect of
clustering their entries, thereby resulting in more frequent draws with no top tier
winners than would otherwise be expected. We illustrate this effect in Figure 19,
35which plots the number of draws on the vertical axis against the aggregate number
of top tier winners on the horizontal axis. The bars indicate the expected
frequencies assuming the binomial distribution in Equation (21) without
conscious selection, whereas the markers represent the observed frequencies.
In order to assess whether the observed frequencies of draws with no top
tier winners are what one would expect given the tendency of players to cluster in
particular combinations of numbers, thereby providing evidence for or against
randomness, we need to take account of conscious selection. We are able to do
this by exploiting information on the coverage rate for each draw. This is the
proportion of combinations selected by at least one player and is sometimes
expressed as a percentage.
The coverage values ci are available for all i = 1,2, K , D draws from 13th
February 2004 to 8th February 2008 inclusive, where D = 209 as before. From
this information, we can calculate the expected number of draws that should
produce at least one top tier winner as
D
exp .(≥ 1) = ∑ ci (22)
i =1
and the expected number of draws that should produce no top tier winners as
D D
exp .(0) = ∑ (1 − ci ) = D − ∑ ci . (23)
i =1 i =1
Consequently, we can perform a chi-square goodness-of-fit test to assess whether
our observed frequencies of jackpot winners agree with the expected frequencies
by chance alone. As there are only two categories, we incorporate a continuity
correction for improved accuracy and the test statistic becomes
362
1
obs. − exp. −
T = ∑
2
(24)
exp.
on one degree of freedom. The binomial test does not apply here, as the coverage
varies across draws.
According to the draw history available to us, the expected frequencies for
U.K. EuroMillions coverage only are exp .(0) ≈ 197 ⋅ 5 and exp .(≥ 1) ≈ 11 ⋅ 5
draws, compared with the observed frequencies of obs.(0) = 201 and obs.(≥ 1) = 8
draws respectively. The corresponding chi-square test statistic and p-value are
T ≈ 0 ⋅ 831 and p ≈ 0 ⋅ 362 . As the latter exceeds 0 ⋅ 05 , the test is not significant
at the 5% level. We conclude that the number of draws generating at least one
U.K. jackpot winner is consistent with mathematical expectation.
Similarly, the expected frequencies for the aggregate EuroMillions coverage
of all participating countries are exp .(0) ≈ 139 ⋅ 4 and exp .(≥ 1) ≈ 69 ⋅ 6 draws,
compared with the observed frequencies of obs.(0 ) = 145 and obs.(≥ 1) = 64
draws respectively. The corresponding chi-square test statistic and p-value are
T ≈ 0 ⋅ 564 and p ≈ 0 ⋅ 452 . As the latter exceeds 0 ⋅ 05 , the test is not significant
at the 5% level. We conclude that the number of draws generating at least one
jackpot winner across all participating countries is consistent with mathematical
expectation.
8. Conclusions
The National Lottery Commission invited the Centre for the Study of
Gambling (University of Salford) to establish whether there are any elements of
non-randomness within EuroMillions. Concentrating on the first four years of
37EuroMillions draws, from 13th February 2004 to 8th February 2008 inclusive,
specific objectives were to test whether:
(a) there is equality of frequency for each EuroMillions number drawn;
(b) each EuroMillions draw is independent of preceding draws;
(c) there is any bias for or against tickets sold in particular countries;
(d) the frequency of top tier winners is what would be expected from
random draws.
Players in any particular draw select five main numbers from 1 to 50 and
two “Lucky Star” numbers from 1 to 9. They may win various prizes by matching
their chosen numbers with selected combinations drawn. 50% of ticket sales are
allocated to the Common Prize Fund, which is distributed among different prize
tiers according to pre-specified percentages. There are no fixed prize tiers. We
present the results of our investigations in this report, beginning with an
exploratory data analysis in Section 2. The findings here are that there appear to
be no obvious discrepancies and all entries in the database appear to be correct.
In Section 3, we perform two important significance tests. One tests
whether the frequencies of the five main numbers drawn agree with the null
hypothesis that the five main balls are drawn at random and another tests whether
the frequencies of the Lucky Star numbers drawn agree with the null hypothesis
that the Lucky Star balls are drawn at random. Neither of our modified chi-square
goodness-of-fit tests is significant at the 5% level, providing evidence in support
of random EuroMillions drawings.
Section 4 presents a set of tests for the sequential independence of
EuroMillions draws. Based on the gaps between successive selections of the
specific main numbers 1 to 50, we employ chi-square goodness-of-fit tests to
compare observed and expected frequencies of possible gap sizes. Using the
Bonferroni method for multiple comparisons to adjust the critical p-value and
38allow for multiplicity, the overall test is not significant at the 5% level and we
conclude that successive EuroMillions main number draws appear to be
independent of one another.
We perform a similar analysis for the nine Lucky Star numbers and this
overall test is not significant at the 5% level, so we conclude that successive
EuroMillions Lucky Star number draws appear to be independent of one another.
As a further check for the possibility of serial dependence, we perform runs tests
for low valued balls and neither of the runs tests for main numbers and Lucky
Stars is significant. This again provides evidence that the outcomes of specific
draws do not depend on preceding draw results.
Section 5 presents graphical comparisons of observed and expected
frequency distributions for the order statistics of the five main numbers and for
the order statistics of the two Lucky Stars. The observed frequencies resemble the
expected frequencies in each case, further supporting our formal tests of
randomness. Then we perform several tests for randomness of the five main
numbers and two Lucky Stars, based upon the means and standard deviations of
the sums of numbers in each draw, and upon the parities of the numbers in each
draw. None of these tests is significant at the 5% level, providing further
evidence in support of random drawings.
Section 6 investigates whether the first four years of EuroMillions draw
history provide any evidence of bias for or against tickets sold in particular
countries. By calculating sample correlation coefficients between the sales
figures for each country and (i) the numbers of jackpot winners, (ii) the total
numbers of winners and (iii) the total prize monies, we are able to measure the
association between these pairs of variables. As illustrated by scatter plots and
regression fits, these correlations are all close to one with no clear outliers,
providing evidence to support the claims of no bias.
39In Section 7, we consider whether the frequency of top tier winners (match
five main numbers plus two Lucky Stars) agrees with expectation under the
assumption that all EuroMillions procedures operate randomly. Knowledge of the
coverage for each draw provides sufficient information to analyse the frequency
of draws with no top tier winners, separately for the U.K. game only and for the
aggregate game across all participating countries. Both chi-square goodness-of-fit
tests show that there are no significant differences between the observed and
expected frequencies. These results provide supporting evidence that the process
of generating winning numbers is independent of the pattern of numbers chosen
by the playing population.
Although we have followed the convention of testing for significance at the
5% level throughout this investigation, it is worth noting that none of the tests we
conducted would be significant even at the 10% level. From all of these results, it
is clear that the first four years of draw history provide no evidence that there are
any elements of non-randomness within EuroMillions.
REFERENCES
Haigh, J. (1997) The statistics of the National Lottery. Journal of the Royal
Statistical Society series A, 160, 187-206.
Joe, H. (1993) Tests of uniformity for sets of lotto numbers. Statistics & Probability
Letters, 16, 181-188.
Miller, I. and Miller, M. (2004) John E. Freund’s Mathematical Statistics with
Applications (seventh edition). Prentice Hall.
Rice, J.A. (2007) Mathematical Statistics and Data Analysis (third edition). Duxbury
Press.
Royal Statistical Society (2000, 2002) Reports on the randomness of the lottery.
National Lottery Commission website: http://www.natlotcomm.gov.uk/
University of Salford (2004, 2005, 2010) Reports on the randomness of lottery
games. National Lottery Commission website: http://www.natlotcomm.gov.uk/
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