A STATISTICAL ANALYSIS OF THE NIGHTTIME 
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 preselect 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 multidraw number from two to 12. Players can win in the following ways:

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

NUMBER  PROBABILITY 
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:
FREQENCY OF OCCURRENCE OF THE 

NUMBER  TIMES OCCURRED 
0  600 
1  616 
2  589 
3  591 
4  589 
5  559 
6  609 
7  597 
8  621 
9  626 
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 Chisquare test is run on the data for the 2000 drawings, the following results are obtained:
X^{2}  =  (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 square^{2}, the Chisquare value needs to be at least 14.68 to indicate nonrandomness with a probability of at least .9, we can not say at this point that the number selections are nonrandom 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.
ORDERED FREQUENCY OF THE 

0  215  0  187  0  198  
1  198  1  192  1  226  
2  187  2  209  2  193  
3  186  3  202  3  203  
4  182  4  210  4  197  
5  186  5  192  5  181  
6  214  6  205  6  190  
7  211  7  201  7  185  
8  209  8  194  8  218  
9  210  9  207  9  209 
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 Chisquare test is run on the data for occurring first in the 2000 drawings, the following results are obtained:
X^{2}  =  (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 Chisquare test is run on the data for occurring second in the 2000 drawings, the following results are obtained:
X^{2}  =  (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 Chisquare test is run on the data for occurring third in the 2000 drawings, the following results are obtained:
X^{2}  =  (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 square^{2}, the Chisquare value needs to be at least 14.68 to indicate nonrandomness with a probability of at least .9. Since each of the Chisquare 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 nonrandom with an error of 10% or less.
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.