Texas LOTTO Lottery Using 54 Balls

A FINAL STATISTICAL ANALYSIS OF THE
TEXAS LOTTO LOTTERY USING 54 BALLS
AFTER 292 DRAWINGS
BEGINNING JULY 19, 2000 AND ENDING MAY 3, 2003

The Rules of the Game

Beginning on July 19, 2000, and ending on May 3, 2003, a machine in Austin, Texas was used to select six balls numbered from 1 to 54 twice a week on Wednesdays and Saturdays. Lottery players attempt to pre-select the winning numbers in order to win various amounts of money. Each Lotto playslip has five places called playboards. Each playboard contains the numbers one through fifty-four. Six numbers can be selected 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 10.

Players can win in the following ways:

match all six of the numbers drawn - odds 1 in 25,827,165,
match five of the six numbers drawn - odds 1 in 89,678,
match four of the six numbers drawn - odds 1 in 1,526, or
match three of the six numbers drawn - odds 1 in 75.
The over-all odds of winning for each play board played are 1 in 57.

Probability of Winning or Losing
The probabilities of the preceding events occurring are calculated as follows. The probability of selecting all six numbers correctly is 1/C(54,6) which is 1/25,827,165 = .000000038. The probability of selecting five correctly is 1/C(54,6) times C(6,5) times C(48,1) which is approximately 1/89,678 = .000011. The probability of selecting four correctly is 1/C(54,6) times C(6,4) times C(48,2) which is approximately 1/1,526 = .000655. Finally, the probability of selecting three correctly is 1/C(54,6) times C(6,3) times C(48,3) which is 1/75 = .0134. The sum of these probabilities is .01406 which is approximately 1/71 - the probability of winning anything.
The probability of losing is also interesting to calculate. The probability that none of the numbers will be chosen is 1/C(54,6) times C(6,0) times C(48,6) which is .475. The probability that exactly one number will be chosen is 1/C(54,6) times C(6,1) times C(48,5) which is .3978. Finally, the probability that exactly two numbers will be chosen is 1/C(54,6) times C(6,2) times C(48,4) which is .113. The probability of losing is the sum of these numbers which is .98594. Of course this number is 1 - .01406, the probability of winning anything at all.
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.
Frequency of Numbers Chosen
The frequencies of each number chosen are shown in the following table:

TABLE 1.

FREQUENCY OF OCCURRENCE OF THE
LOTTO LOTTERY NUMBERS
USING 54 BALLS
AFTER 292 DRAWINGS

NO. TIMES NO. TIMES NO. TIMES

1 26 19 26 37 33

2 34 20 27 38 24

3 37 21 31 39 33

4 42 22 35 40 35

5 38 23 28 41 44

6 27 24 28 42 40

7 24 25 33 43 32

8 23 26 28 44 29

9 41 27 38 45 36

10 34 28 39 46 28

11 38 29 24 47 33

12 33 30 34 48 36

13 28 31 27 49 29

14 27 32 39 50 36

15 33 33 45 51 32

16 35 34 26 52 32

17 25 35 37 53 30

18 36 36 32 54 32

Theoretically, the probability P(x) that any given number x will be one of the six drawn is:

C(1,1)C(53,5)
P(x) = ----------------------- = .111111,
C(54,6)
so the number of times we expect x to occur in n drawings is n times .111111. Since there have been 292 drawings, x should have occurred 292 times .111 or 32.444444 times. Compare this theoretical frequency with the actual frequencies given in the above table.
Mean, Standard Deviation and Distribution of Numbers Chosen
If the machine is choosing the numbers randomly, the average number chosen from the numbers 1 to 54 should be 27.5 and the standard deviation should be 15.73. The actual average number chosen by the machine after 292 drawings is 27.78 and the standard deviation is 15.64.
In order to run the Chi Square Test, the numbers were grouped into nine intervals of six numbers each. Table 2. shows how many numbers occurred in the intervals 1 to 6, 7 to 12, etc. up to the interval 49 to 54.

TABLE 2.

INTERVAL DISTRIBUTION OF THE
LOTTO LOTTERY NUMBERS
USING 54 BALLS
AFTER 292 DRAWINGS

INTERVAL NUMBERS OCCURRING

1 TO 6 204

7 TO 12 193

13 TO 18 184

19 TO 24 175

25 TO 30 196

31 TO 36 206

37 TO 42 209

43 TO 48 194

49 TO 54 191

Since 1,752 numbers have been chosen and there are nine intervals, the average number of numbers in each interval should be 194.66667. The standard deviation is 10.84. Since the intervals are six units in length, the expected number of numbers in each interval after 292 drawings is 32.444444 times 6 or 194.66667 which agrees with the average above.
The Chi-square test can now be run on the data in the intervals for 100 drawings as follows:

X² = (204 - 194.67)²/194.67 + (193 - 194.67)²/194.67 +

(184 - 194.67)²/194.67 + (175 - 194.67)²/194.67 +

(196 - 194.67)²/194.67 + (206 - 194.67)²/194.67 +

(209 - 194.67)²/194.67 + (194 - 194.67)2/⁴194.67 +

(191 - 194.67)²/194.67

= 954.49/194.67 = 4.828771.

Since, according to a table of critical values of Chi square ¹, the Chi square value needs to be at least 13.4 to indicate non-randomness with a probability of at least .9, we can not say that the number selections were non-random with an error of 10% or less.

REFERENCE
1. Mendenhall, William and Beaver, Robert J., Introduction to Probability and Statistics, PWS-Kent Publishing Co., Boston, 1991, pp 670 - 671.
2. Lamb, Jr., John, Huffstutler, Ron, Brock, Archie and Aslan, Bill, "A Statistical Analysis of the Texas Lottery," Texas Mathematics Teacher*, Vol. XLI(1), January, 1994, pp 4 - 17.

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