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A FINAL STATISTICAL ANALYSIS OF THE |
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A FINAL STATISTICAL ANALYSIS OF THE |
The Rules of the Game
Twice a week, on Tuesday and Friday, from October 13, 1995 to November 22, 1996, a machine in Austin, Texas was used to select five balls numbered from 1 to 39. Due to the popularity of the game, the number of drawings per week was increased to four on November 25, 1996. The machine selected five balls on Mondays, Tuesdays, Thursdays and Fridays. This lasted until July 26, 2002 at which time the game was revised to have only 37 balls and to allow matching two balls to win $2
Lottery players attempt to pre-select the winning numbers to be awarded various amounts of money. Each Lotto playslip has five places called playboards. Each playboard contains the numbers one through thirty-nine. Five numbers can be selected on any or all 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 ten. On the playslip, it says players can win in the following ways:
The over-all odds of winning for each play board played are 1 in 100.
Probability of Winning or Losing
The probabilities of the preceding events occurring are calculated as follows. The probability of selecting all five numbers correctly is 1/C(39,5) times [C(5,5) times C(34,0)] which is 1/575,757 which is approximately .00000174. The probability of selecting four correctly is 1/C(39,5) times [C(5,4) times C(34,1)] which is 170/575,757 which is approximately .0002952. Finally, the probability of selecting three correctly is 1/C(39,5) times [C(5,3) times C(34,2)] which is 5,610/575,757 which is approximately = .0097087. The sum of these probabilities is approximately .0100213 which is approximately 1/100 -- 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(39,5) times [C(5,0) times C(34,5)] which is approximately .4832872. The probability that exactly one number will be chosen is 1/C(39,5) times [C(5,1) times C(34,4)] which is approximately .4027394. Finally, the probability that exactly two numbers will be chosen is 1/C(39,5) times [C(5,2) times C(34,3)] which is approximately .1039327. The probability of losing is the sum of these numbers which is approximately .9899593. Of course this number is 1 - .0100407, the probability of winning anything at all.
Odds Versus Probability
On the playslips, it states the odds of winning. However, as we have seen above, the numbers actually printed on the playslips are the probabilities of winning. These numbers are usually quite different. If p is the probability of winning an event, then 1 - p is the probability of losing that event. The odds of winning that event are the probability of winning divided by the probability of losing or p/(1 - p). Suppose the probability of winning an event were 1/3. Then the probability of losing the event would be 2/3, so the odds of winning that event would be (1/2)/(2/3) = 1/2 which is quite different from the probability of winning. Fortunately, when the probabilities for winning an event are very small, the probabilities and odds are very close to the same number. In the case of the CASH 5 Lottery, we have the following:
ODDS VERSUS PROBABILITIES |
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| ODDS | PROBABILITY | DIFFERENCE | |
| Matching 5 | 0.000001736846 | 0.000001736843 | 0.000000000003 |
| Matching 4 | 0.000295350659 | 0.000295263453 | 0.000000087206 |
| Matching 3 | 0.009839567957 | 0.009743694216 | 0.000095873741 |
The difference column would seem to indicate that there would be no problem using the terms odds and probabilities interchangeably when discussing the CASH 5 Lottery.
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
Theoretically, the probability P(x) that any given number x will be one of the five drawn is:
which is the hypergeometric probability formula. So the number of times we expect x to occur in n drawings is n times .13. Since there have been 1301 drawings at the time of this writing, x should have occurred 1301 multiplied by .13 times or 169 times. Compare this theoretical frequency with the actual frequencies given in the following table:
FREQUENCY OF OCCURRENCE OF |
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| 1 | 157 | 14 | 165 | 27 | 164 | ||
| 2 | 176 | 15 | 186 | 28 | 161 | ||
| 3 | 166 | 16 | 185 | 29 | 158 | ||
| 4 | 150 | 17 | 172 | 30 | 144 | ||
| 5 | 161 | 18 | 154 | 31 | 163 | ||
| 6 | 168 | 19 | 173 | 32 | 166 | ||
| 7 | 174 | 20 | 178 | 33 | 160 | ||
| 8 | 173 | 21 | 174 | 34 | 168 | ||
| 9 | 166 | 22 | 177 | 35 | 155 | ||
| 10 | 151 | 23 | 158 | 36 | 177 | ||
| 11 | 166 | 24 | 159 | 37 | 191 | ||
| 12 | 171 | 25 | 173 | 38 | 159 | ||
| 13 | 170 | 26 | 169 | 39 | 167 | ||
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 39 should be 20 and the standard deviation should be 11.4. The actual average number chosen by the Texas Cash 5 Lottery machine over 1301 drawings is 19.992 and the actual standard deviation is 11.2244.
For the Chi-square test, the 39 CASH 5 numbers were grouped into 13 intervals containing three numbers each as follows: 1, 2 and 3; 4, 5 and 6; 7, 8 and 9; . . . ; 37, 38 and 39. The following table shows the total number of times the three numbers occurred in their respective intervals:
CASH 5 NUMBER DISTRIBUTION |
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| 1 to 3 | 499 |
| 4 to 6 | 479 |
| 7 to 9 | 513 |
| 10 to 12 | 488 |
| 13 to 15 | 521 |
| 16 to 18 | 511 |
| 19 to 21 | 525 |
| 22 to 24 | 494 |
| 25 to 27 | 506 |
| 28 to 30 | 463 |
| 31 to 33 | 489 |
| 34 to 36 | 500 |
| 37 to 39 | 517 |
Because 6,505 numbers have been chosen so far and there are thirteen intervals, the average number of numbers in each interval is 6,505/13 = 501. Note that this number is also three times the expected occurrence of each number found earlier, (5/39) times 1301. The Chi-square test can now be run on the data in the intervals for 1301 drawings as follows:
| X2 | = | (499 - 501)2/501 + (479 - 501)2/501 + |
| (513 - 501)2/501 + (488 - 501)2/501 + | ||
| (521 - 501)2/501 + (511 - 501)2/501 + | ||
| (525 - 501)2/501 + (494 - 501)2/501 + | ||
| (506 - 501)2/501 + (463 - 501)2/501 + | ||
| (489 - 501)2/501 + (500 - 501)2/501 + | ||
| (517 - 501)2/501 | ||
| = | 7.58. |
According to a table of critical values of Chi square1, the Chi square value needs to be at least 18.55 to indicate non-randomness with a probability of at least .9, so it cannot be concluded that after 1301 drawings, the number selections are non-random with an error of 10% or less.
Mathematical Expectation
The CASH 5 prizes are awarded by dividing the total ticket sales for each drawing by two. Half goes to the state and half goes to the winners. This latter half, call it T, is divided among the winners as follows: 20% of T is distributed among those who matched 5 numbers, 30% of T is distributed among those who matched 4 numbers and 50% of T is distributed among those who matched 3 numbers. If X people matched all 5 numbers, then each of them win .2T/X dollars. If Y people matched 4, then they each win .3T/Y. Finally, if Z people matched 3, then they win .5T/Z. There is one exception to these rules: in case no one matches 5 numbers, the prize for doing that is added to the one for matching 4.
The empirical probabilities of matching 5, 4 and 3 numbers are X/2T, Y/2T and Z/2T where each is the number of winners divided by the total number of players. Since it costs $1 to play the game, it takes 2T players to generate T dollars for prizes since the state takes 50% off the top.
The mathematical expectation for an event is the product of the probability of that event times the value of the prize for winning the event. For the CASH 5 Lottery mathematical expectation, we have the following table:
| Probability | Prize | Product |
| X/2T | .2T/X | .2T/2T |
| Y/2T | .3T/Y | .3T/2T |
| Z/2T | .5T/Z | .5T/2T |
Since .2T + .3T + .5T = T, the sum if these expectations is T/2T = 1/2. Thus, a player can expect to win 50 cents for every dollar spent on a ticket.
Conclusions:
It was interesting to keep track of the number behavior for the Texas CASH 5 Lottery using 39 numbers for its 1301 drawings. There is no evidence of non-randomness in the number selection process and players can expect to win about 50 cents on the dollar since the state takes 50% off the top before awarding prizes.
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 F., Huffstutler, Ron, Brock, Archie and Aslan, Farhad (Bill). "A Statistical Analysis of the Texas Lottery," Texas Mathematics Teacher, Vol. XLI (1) January, 1994.
3. Lamb, Jr., John F., "A Statistical Analysis of the Texas Cash 5 Lottery after 350 Drawings," Texas Mathematics Teacher, Vol. XLVI (1) Spring, 1999.
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