Texas LOTTO Lottery Using 50 Balls

A FINAL STATISTICAL ANALYSIS OF THE
TEXAS LOTTO LOTTERY USING 50 BALLS
AFTER 805 DRAWINGS
FROM NOVEMBER 19, 1992 TO JULY 15, 2000

Introduction
The first Texas Lotto lottery began on November 14, 1992 using 50 balls numbered from 1 to 50. Due to declining sales of tickets, the Texas Lottery Commission decided to increase the number of balls to 54 in hopes of offering bigger jackpots to encourage more ticket sales. Thus, the last Lotto drawing using 50 balls was held on July 15, 2000. The first drawing using 54 balls occurred on July 19, 2000. This article is an analysis of the overall behavior of the numbers drawn for the Texas Lotto Lottery using 50 balls.
The Rules of the Game
A machine in Austin, Texas was used to select six balls numbered from 1 to 50 twice a week on Wednesdays and Saturdays. Lottery players attempted to pre-select the winning numbers in order to win various amounts of money. Each Lotto playslip had five places called playboards. Each playboard contained the numbers one through fifty. Six numbers could be selected for any or all of the playboards. Provision was made for these numbers to be entered into more than one drawing by marking a multi-draw number from two to 10. Players could win in the following ways:

match all six of the numbers drawn - odds 1 in 15,890,700,
match five of the six numbers drawn - odds 1 in 60,192,
match four of the six numbers drawn - odds 1 in 1,120, or
match three of the six numbers drawn - odds 1 in 60.
The over-all odds of winning for each play board played were 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(50,6) which is 1/15,890,700 = .000000063. The probability of selecting five correctly is 1/C(50,6) times C(6,5) times C(44,1) which is approximately 1/60,192 = .000016. The probability of selecting four correctly is 1/C(50,6) times C(6,4) times C(44,2) which is approximately 1/1,120 = .0009. Finally, the probability of selecting three correctly is 1/C(50,6) times C(6,3) times C(44,3) which is 1/60 = .0167. The sum of these probabilities is .0176 which is approximately 1/57 - 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(50,6) times C(6,0) times C(44,6) which is .444. The probability that exactly one number will be chosen is 1/C(50,6) times C(6,1) times C(44,5) which is .41. Finally, the probability that exactly two numbers will be chosen is 1/C(50,6) times C(6,2) times C(44,4) which is .128. The probability of losing is the sum of these numbers which is .9824. Of course this number is 1 - .0176, 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
FIFTY LOTTO LOTTERY NUMBERS
AFTER 805 DRAWINGS

NO. TIMES NO. TIMES NO. TIMES NO. TIMES

1 109 14 92 27 98 40 99

2 92 15 99 28 104 41 107

3 104 16 104 29 104 42 85

5 98 18 81 31 110 44 95

6 88 19 103 32 108 45 101

7 94 20 97 33 104 46 104

8 98 21 88 34 94 47 89

9 100 22 103 35 86 48 92

10 104 23 85 36 90 49 100

11 99 24 94 37 75 50 92

12 96 25 83 38 99

13 91 26 103 39 107

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

C(1,1)C(49,5)
P(x) = ----------------- = .12,
C(50,6)
so the number of times we expect x to occur in n drawings is n times .12. Since there were 805 drawings, x should have occurred 805 times .12 or 96.6 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 50 should be 25.5 and the standard deviation should be 14.577. The actual average number chosen by the machine in 805 drawings was 25.47 and the standard deviation was 14.50.
In order to run the Chi Square Test, the numbers were grouped into ten intervals of five numbers each. Table 2. shows how many numbers occurred in the intervals 1 to 5, 6 to 10, etc. up to the interval 46 to 50.

TABLE 2.

INTERVAL DISTRIBUTION OF THE
LOTTO LOTTERY NUMBERS
AFTER 805 DRAWINGS

INTERVAL NUMBERS OCCURRING

1 TO 5 503

6 TO 10 484

11 TO 15 477

16 TO 20 478

21 TO 25 453

26 TO 30 495

31 TO 35 502

36 TO 40 470

41 TO 45 487

46 TO 50 481

Since 4830 numbers were chosen there are ten intervals, the average number of numbers in each interval should be 483. The standard deviation is 15.11. Since the intervals are five units in length, the expected number of numbers in each interval after 805 drawings is 96.6 times 5 or 483 which agrees with the average above.
The Chi-square test can now be run on the data in the intervals for 805 drawings as follows:

X² = (503 - 483)²/483 + (484 - 483)²/483 +

(477 - 483)²/483 + (478 - 483)²/483 +

(453 - 483)²/483 + (495 - 483)²/483 +

(502 - 483)²/483 + (470 - 483)2/⁴483 +

(487 - 483)²/483 + (481 - 483)²/483

= 2056/483 = 4.26.

Since, according to a table of critical values of Chi square ¹, the Chi square value needs to be at least 14.7 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.
Mathematical Expectation
Now that 805 sets of six numbers have been drawn, one might ask, based on past performance, how much can a player expect to win on a single play of the game. Since there are four ways to win, one needs to calculate the amount of winnings for each of them. Each drawing begins with a prize of approximately four million dollar for matching all six numbers. If there are no winners, the amount of the prize increases to approximately ten million dollars, then to approximately $17 million if no winner is drawn and so on until someone wins. If there are multiple winners, the prize is shared equally. The prize for matching three numbers remains constant at three dollars (d₃ = 3). The prize for matching four numbers has varied from a low of $34 (d_4L = 34) to a high of $248 (d_4H = 248) with an average of $107 (d_4A = 107).
The prize for matching five numbers has varied from a low of $424 (d_5L = 424) to a high of $6,923 (d_5H = 6,923) with an average of $1,741 (d_5A = 1,741).
The beginning prize value for matching six numbers is approximately $4,000,000 (d₆ = 4,000,000).
If one multiplies the prize values d_ij times their respective probabilities p_k computed earlier, we obtain the mathematical expectation, d_ij times p_k for each. The following table shows the results:

Table 3.

MATHEMATICAL EXPECTATION
FOR MATCHING THREE OF SIX

FIXED PRIZE (d₃)(p₃) = (3/60) = $.05

MATHEMATICAL EXPECTATION
FOR MATCHING FOUR OF SIX

EXPECTED HIGH (d_4H)(p₄) = (248/1120) = $.22

EXPECTED AVERAGE (d_4A)(p₄) = (107/1120) = $.10

EXPECTED LOW (d_4L)(p₄) = (34/1120) = $.03

MATHEMATICAL EXPECTATION
FOR MATCHING FIVE OF SIX

EXPECTED HIGH (d_5H)(p₅) = (6923/60192) = $.12

EXPECTED AVERAGE (d_5A)(p₅) = (1741/60192) = $.03

EXPECTED LOW (d_5L)(p₅) = (424/60192) = $.01

MATHEMATICAL EXPECTATION
FOR MATCHING SIX OF SIX

$4 MILLION PRIZE (d₆)(p₆) = (4000000/15890700) = $.25

At the $4 million level, the most a player can expect to win is $.05 + $.22 + $.12 + $.25 = $.64 and the least a player can expect to win is $.05 + $.03 + $.01 + $.25 = $.34. The average amount a player can expect to win is $.05 + $.10 + $.03 + $.25 = $.43. Since the cost of playing the LOTTO Lottery is one dollar, one can expect to lose $.57 on the average each time the LOTTO lottery is played at the $4 million level.
However, the prize for matching six numbers grows until someone wins, so the logical question is; "How large should this prize be in order to assure the expectation of winning a dollar or more. If x is this amount, then:

$1.00 = $.05 + $.10 + $.03 + $ x / 15,890,700.

Thus,

$.82 = $ x / 15,890,700

so,

x = $(.82)(15,890,700) = $13,030,374.
That is, when the prize for matching six numbers exceeds $13,030,374, one can expect to win more than it costs to play the game.
Of course, this analysis assumes there is only one person who matched all six numbers. However, as the prize grows and more people play the game, the likelihood of multiple winners grows as well. Another study might be made to determine the range of values for the big prize so that a player could expect to win more than it costs to play the game.
Note that this expectation can never be a certainty since there have been multiple winners at the $4 million level and single winners at the $45 million level.

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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