RANDOMNESS OF THE LOTTO DRAWS: SUMMARY OF FINDINGS

Centre for the Study of Gambling
University of Salford

Greater Manchester, M5 4WT, United Kingdom

RANDOMNESS OF THE
LOTTO DRAWS:

SUMMARY OF FINDINGS

A Report for the
National Lottery Commission

June 2004

Randomness of the Lotto Draws:
Executive Summary

ABSTRACT
A variety of hypothesis tests for randomness of the Lotto draws up to and including
10th April 2004 was conducted. These cover aspects relating to the frequencies of
numbers drawn, occurrences of rollovers and double rollovers, possible
dependencies between draws and combinations of ball sets and draw machines. All
of these tests support the hypothesis of randomness. That is, they confirm no
evidence of non-randomness amongst the Lotto draws.

1. Introduction

As requested by the National Lottery Commission, the Centre for the Study
of Gambling at the University of Salford checked whether there are any elements
of non-randomness within the UK National Lottery. Concentrating on the
Saturday and Wednesday Lotto draws, specific objectives were to test whether:

(a) there is equality of frequency for each Lotto number drawn;

(b) Lotto draws are independent of preceding draws;

(c) the frequency of rollovers and double rollovers is as expected;

(d) there is any evidence of non-randomness in any combinations of
ball sets and Lotto draw machines.

1

We considered all 870 draws up to and including 10th April 2004,
comprising draw numbers 1 to 866 and four bonus draws. Three of these took
place on 18th May 2002, between draws 668 and 669, and the fourth took place on
1st June 2002, between draws 672 and 673. The full results of our investigations
are presented in the report “Randomness of the Lotto Draws: Detailed Findings”.
This report summarizes those results.

First we undertook an exploratory analysis of the data in order to check that
there were no unusual, suspicious or incorrect observations. Then we performed a
range of suitable hypothesis tests for randomness to answer the above questions.
The sample size is 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% level of significance, corresponding
with a 95% confidence level, thereby maintaining consistency with most
published statistical analyses and ensuring ease of interpretation.

2. Exploratory Data Analysis

The table and bar chart below present the observed frequencies of
occurrence (f) for each of the numbers (n) of the six main balls selected during the
first 870 draws up to and including 10th April 2004.

2

f requenc ynfnfnfnfnfnfnf
1 104 8 101 15 98 22 98 29 104 36 100 43 121
2 114 9 110 16 98 23 120 30 111 37 97 44 125
3 101 10 109 17 102 24 103 31 116 38 137 45 113
4 97 11 115 18 109 25 123 32 117 39 97 46 112
5 101 12 107 19 108 26 103 33 106 40 113 47 121
6 107 13 87 20 87 27 108 34 102 41 88 48 115
7 108 14 99 21 93 28 113 35 110 42 97 49 95

140
120
100

80
60
40
20

0
0 10 20 30 40 50

number

The next table and bar chart present the observed frequencies of occurrence
(f) for each of the numbers (n) of the bonus balls selected during the first 870
draws up to and including 10th April 2004.

3

f requenc ynfnfnfnfnfnfnf
1 16 8 19 15 16 22 23 29 21 36 20 43 17
2 14 9 25 16 16 23 18 30 19 37 23 44 14
3 18 10 19 17 18 24 17 31 24 38 24 45 19
4 15 11 19 18 13 25 23 32 12 39 11 46 18
5 18 12 16 19 7 26 19 33 22 40 11 47 19
6 19 13 11 20 23 27 13 34 17 41 17 48 14
7 14 14 20 21 24 28 20 35 16 42 18 49 17

25
20
15
10

5
0

0 10 20 30 40 50

number

Regarding the machines used to generate the Lotto numbers, the frequency
table below shows that a total of ten machines were used during the first 870
draws. Clearly, there is plenty of variability built into the process to help avoid
any systematic bias that might be inherently present in any particular machine.

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Machine First draw Last draw Total count
appearance appearance
Guinevere 192
Arthur 1 754 237
Lancelot 2 816 182
Merlin 20 817 110
Galahad 268 713 52
Vyvyan 506 810 48
512 705 22
Amethyst 818 866 18
Moonstone 819 865
824 832 3
Opal 834 858 6
Topaz

Similarly, the frequency table below shows that a total of twenty-two ball
sets were used during the first 870 draws. Again, there is plenty of variability
built into the process to help avoid any systematic bias that might be inherently
present in any ball set.

Ball set Total count Ball set Total count
A 2 11 21
B 6 12 19
1 67 14 21
2 81 L1 21
3 91 L2 17
4 98 L3 19
5 79 L4 18
6 93 L5 16
7 95 L6 18
8 52 L7 16
10 4 L8 16

5

The histogram below displays the sales per draw (excluding the four bonus
draws) in millions of pounds.

200

f requenc y 100

0 20 40 60 80 100 120 140
0
sales (£m)

The histogram below displays the coverage rates for available draws,
expressed as percentages. There is a clear mixture of two distinct distributions,
which correspond with Wednesday and Saturday draws. Notice that, amongst
these draws, there are no coverage rates below 69%. Although some
combinations of numbers are more popular than others, the very large numbers of
players choose sufficiently differently from each other to ensure that well over
half of all possible combinations are chosen by at least one person in each draw.
The Lucky Dip facility, where numbers are chosen for the purpose of the ticket,
was introduced from draw 71 and further helps spread the players’ choices across
a large proportion of the 13,983,816 possible combinations.

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f requenc y 70
60
50
40
30
20
10

0
0 10 20 30 40 50 60 70 80 90 100

coverage (%)

3. Testing whether there is Equality of Frequency
for each Lotto Number Drawn

Perhaps the most important test is whether the observed frequencies of the
forty-nine Lotto numbers accord with random drawings. We test the equality of
marginal frequencies for the six main numbers chosen in Lotto draws using a
modified goodness-of-fit test. This test is not significant at the 5% level and so
we conclude that there is no evidence of any bias in the selection of particular
numbers in the National Lottery draws.

Next, we consider the bonus balls and test whether all numbers are equally
likely to be chosen as the bonus ball using a standard chi-square goodness-of-fit
test. This test is not significant at the 5% level and so we conclude that there is no
evidence to suggest that the selection of the bonus ball is anything but random.

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4. Testing whether Lotto Draws are Independent
of Preceding Draws

Now, we perform a set of tests for the sequential independence of Lotto
draws. This involves comparing the observed gap frequencies between successive
appearances of all forty-nine numbers with those expected if Lotto draws are
independent of one another. Allowing for multiple comparisons, the standard
goodness-of-fit tests employed are all not significant at the 5% level. We
conclude that there is no evidence of dependence between Lotto draws.

5. Other Tests for Randomness of the Six Main
Lotto Numbers Drawn

We present three further tests for randomness of the six main Lotto numbers
drawn. The first and second are based upon whether the sums of the six main
numbers drawn agree, both in terms of average and spread, with what would be
expected if numbers were drawn at random. For the Lotto draw history, neither of
these tests is significant at the 5% level, again providing evidence in favour of
randomness in the Lotto draws.

The third test is based upon the observed frequencies of odd and even
combinations that occur in given Lotto draws. Using a chi-square goodness-of-fit
test to compare these with the expected frequencies, our result is not significant at
the 5% level, again supporting the hypothesis of random Lotto draws.

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6. Testing whether the Frequency of Rollovers
and Double Rollovers is as Expected

We consider whether the observed frequencies of rollovers and double
rollovers agree with what would be expected if there were no improper
intervention in Lotto procedures. The process of assessment here is not
straightforward. However, knowledge of the coverage for each draw provides
sufficient information to answer the questions posed. Both tests provide
supporting evidence that the process of generating numbers was independent of
the pattern of numbers chosen by the playing population.

The sample mean coverage for all draws is 84 ⋅ 79% , so the expected
proportion of draws that should produce rollovers is 0 ⋅1521. Overall, the sample
proportion of draws that produced no winners is 135 ÷ 866 = 0 ⋅1559 . Using a
suitable test that the true proportion of draws that result in rollovers is 0 ⋅1521
based on our sample proportion, the result is not significant at the 5% level and so
we conclude that the number of rollovers that have actually occurred is in
accordance with what would be expected by chance.

Similarly, there were 12 double rollovers in the first 866 draws (excluding
the bonus draws) and no triple rollovers. As identified above, the expected
proportion of all draws that result in no winners is 0 ⋅1521. Hence, the
probability of a double rollover is estimated to be 0 ⋅1521× 0 ⋅1521 = 0 ⋅ 0231. We
test whether our observed frequency of double rollovers accords with this
expected probability using a suitable hypothesis test. Again, the test is not
significant and we conclude that the observed frequency of double rollovers
accords with what would be expected under random drawings.

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7. Testing whether there is any Evidence of Non-
randomness in any Combinations of Ball Sets
and Lotto Draw Machines

Finally, we consider whether there is any evidence of non-randomness in
any combinations of ball sets and Lotto draw machines. We test the equality of
frequencies of the six main numbers drawn, for all draw machines and ball sets
that have been used, by means of modified goodness-of-fit tests. Allowing for
multiple comparisons, none of our results is significant. We conclude that there is
no evidence to suggest that any particular draw machines or ball sets result in non-
randomness of the six main lottery numbers drawn.

8. Conclusion

A variety of hypothesis tests for randomness of the Lotto draws up to and
including 10th April 2004 was conducted. These cover aspects relating to the
frequencies of numbers drawn, occurrences of rollovers and double rollovers,
possible dependencies between draws and combinations of ball sets and draw
machines. All of these tests support the hypothesis of randomness. That is, they
confirm no evidence of non-randomness amongst the Lotto draws.

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