A STATISTICAL ANALYSIS OF THE NIGHTTIME
TEXAS PICK 3
LOTTERY

AFTER 2,000 DRAWINGS

The Rules of the Pick 3 Night and Day Lottery Game

Currently, twice a day, six times a week, a machine is used to select three balls numbered from 0 to 9 with repetitions allowed. Pick 3 Lottery players attempt to pre-select the winning numbers in order to win various amounts of money.

Each Pick 3 playslip has five places called playboards. Each playboard contains the numbers zero through nine in three columns so that three numbers from zero to nine can be selected in order for any or all of the playboards. Provision is made for these numbers to be entered into more than one drawing by marking a multi-draw number from two to 12. Players can win in the following ways:

  • Match all three of the numbers drawn in the same order - odds 1 in 1,000 - and win $500 if you paid $1 to play or $250 if you paid 50 cents to play.

  • Choose three numbers with two alike and match all three of the numbers drawn in any order - odds 1 in 333 - and win $160 if you paid $1 to play or $80 if you paid 50 cents to play.

  • Choose three different numbers and match all three of the numbers drawn in any order - odds 1 in 167 - and win $80 if you paid $1 to play or $40 if you paid 50 cents to play.

It is interesting at this point to note the mathematical expectation for each of the three ways to win if we pay $1 to play. For the first way, we can expect to win $500 times .001 or 50 cents. For the second way, we can expect to win $160 times 1/333 or 48 cents. For the third way, we can expect to win $80 times 1/167 or 48 cents. The mathematical expectations for each of the three ways to win when 50 cents is paid to play are 25 cents, 24 cents and 24 cents respectively.

Probability of Winning

The probabilities of the preceding events occurring are calculated as follows. The probability of selecting all three numbers in the same order correctly is 1/10 times 1/10 times 1/10 which is 1/1,000. The probability of selecting three in any order correctly when two numbers are alike is 3/1000 since there are 3!/2! = 3 ways to arrange the three numbers. The probability of selecting three in any order correctly when the numbers are all different is 6/1000 since there are 3! = 6 ways to arrange the three numbers. This information is printed on the back of each Pick 3 playslip as the odds of winning.

Recall the odds of winning are the probability of winning divided by the probability of losing, so the odds of selecting all three numbers correctly in the same order would be 1/1000 divided by 999/1000 which is 1/999. However, since the denominators of the probabilities above are fairly large, the odds of winning are approximately the same as the probability of winning because 1/1000 = .001 and 1/999 = .001001. These differ by only .000001.

Randomness of the Lottery

The most important property of any lottery is that the numbers be chosen randomly. In order to test the Lotto numbers, the following measures were used: frequency of the numbers chosen, the mean, standard deviation and the Chi square test.

Probability and Frequency of Numbers Chosen

Theoretically, the probability P(x) that any given number x will be one of the three numbers chosen from the set of ten is one minus the probability of not being chosen at all. The probability that a number will not be drawn is 9/10 which is one minus the probability that it will. The probability that a number will not be chosen when three numbers are drawn is 9/10 times 9/10 times 9/10. Thus the probability P(x) that x will occur is:

P(x) = 1-(9/10)(9/10)(9/10) = 1-.729 = .271

Compare this theoretical probability with the actual probabilities of each number computed from the number of times it has occurred in the 2000 drawings.

TABLE 1.

EMPIRICAL PROBABILITY OF THE
TEN PICK 3 LOTTERY NUMBERS
AFTER 2000 DRAWINGS

NUMBERPROBABILITY
0.271
1.27746
2.26653
3.26734
4.26653
5.25426
6.27463
7.26978
8.27947
9.28147

Since repetitions are allowed in the Pick 3 lottery and three numbers are chosen, the probability that a number will occur at least once is 1/10 + 1/10 + 1/10 = 3/10. Since there have been 2000 drawings at the time of this writing, theoretically, a number should have occurred 3/10 times 2000 or 600 times. Compare this theoretical frequency with the actual frequencies for each number in 2000 drawings:

TABLE 2.

FREQENCY OF OCCURRENCE OF THE
TEN PICK 3 LOTTERY NUMBERS
AFTER 2000 DRAWINGS

NUMBERTIMES OCCURRED
0600
1616
2589
3591
4589
5559
6609
7597
8621
9626

Mean, Standard Deviation and Distribution of Numbers Chosen

If the machine is choosing the numbers randomly, the average number chosen from the numbers 0 to 9 should be 4.5 and the standard deviation should be 2.87. The actual average number chosen by the Texas Pick 3 Lottery machine in 2000 drawings is 4.5255 and the standard deviation is 2.8978.

When the Chi-square test is run on the data for the 2000 drawings, the following results are obtained:

X2=(600 - 600)2/600 + (616 - 600)2/600 +
(589 - 600)2/600 + (591 - 600)2/600 +
(589 - 600)2/600 + (559 - 600)2/600 +
(609 - 600)2/600 + (597 - 600)2/600 +
(621 - 600)2/600 + (626 - 600)2/600
=5.778.

Since, according to a table of critical values of Chi- square2, the Chi-square value needs to be at least 14.68 to indicate non-randomness with a probability of at least .9, we can not say at this point that the number selections are non-random with an error of 10% or less.

Since the Pick 3 Lottery involves the order in which the numbers are chosen, the ordered frequency of the ten Pick 3 numbers was tabulated for the 2000 drawings. The following table shows how often the numbers occurred first, second and third.

TABLE 3

ORDERED FREQUENCY OF THE
TEN PICK 3 LOTTERY NUMBERS
AFTER 2000 DRAWINGS

FIRST
SECOND
THIRD
021501870198
119811921226
218722092193
318632023203
418242104197
518651925181
621462056190
721172017185
820981948218
921092079209

Theoretically, each number should occur 200 times in 2000 drawings since the probability of occurrence at a specified position is 1/10. Compare this theoretical frequency with the actual frequencies in the table above.

When the Chi-square test is run on the data for occurring first in the 2000 drawings, the following results are obtained:

X2=(215 - 200)2/200 + (198 - 200)2/200 +
(187 - 200)2/200 + (186 - 200)2/200 +
(182 - 200)2/200 + (186 - 200)2/200 +
(214 - 200)2/200 + (211 - 200)2/200 +
(209 - 200)2/200 + (210 - 200)2/200
=8.06.

When the Chi-square test is run on the data for occurring second in the 2000 drawings, the following results are obtained:

X2=(187 - 200)2/200 + (192 - 200)2/200 +
(209 - 200)2/200 + (202 - 200)2/200 +
(210 - 200)2/200 + (192 - 200)2/200 +
(205 - 200)2/200 + (201 - 200)2/200 +
(194 - 200)2/200 + (207 - 200)2/200
=2.965.

When the Chi-square test is run on the data for occurring third in the 2000 drawings, the following results are obtained:

X2=(198 - 200)2/200 + (226 - 200)2/200 +
(193 - 200)2/200 + (203 - 200)2/200 +
(197 - 200)2/200 + (181 - 200)2/200 +
(190 - 200)2/200 + (185 - 200)2/200 +
(218 - 200)2/200 + (209 - 200)2/200
=9.19.

Again, according to a table of critical values of Chi- square2, the Chi-square value needs to be at least 14.68 to indicate non-randomness with a probability of at least .9. Since each of the Chi-square values above are less than 14.68, we can not say at this point that the number selections in first, second or third places are non-random with an error of 10% or less.

REFERENCES

1. Lamb, John, Huffstutler, Ron, Brock, Archie and Aslan, Farhad, "A Statistical Analysis of the Texas Lottery," Texas Mathematics Teacher, January, 1994.

2. Mendenhall, William and Beaver, Robert J., Introduction to Probability and Statistics, PWS- Kent Publishing Co., Boston, 1991, pp 670 - 671.


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