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A STATISTICAL ANALYSIS OF THE |
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A STATISTICAL ANALYSIS OF THE |
On July 29, 2002 the Texas Cash 5 Lottery began drawings six times a week, Monday through Saturday, using only 37 balls and allowing the matching of two numbers to win $2.00.
These changes give the game a whole new perspective.
The Rules of the Game
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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-seven. 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 8. |
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(37,5) times [C(5,5) times C(32,0)] which is 1/435,897 which is approximately .000002294. The probability of selecting four correctly is 1/C(37,5) times [C(5,4) times C(32,1)] which is 160/435,897 which is approximately .000367. The probability of selecting three correctly is 1/C(37,5) times [C(5,3) times C(32,2)] which is 4,960/435,897 which is approximately = .01138. Finally, the probability of selecting two correctly is 1/C(37,5) times [C(5,2) times C(32,3)] which is 49,600/435,897 which is approximately = .1138. The sum of these probabilities is approximately .12555 which is approximately 1/8 -- 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(37,5) times [C(5,0) times C(32,5)] which is 201,376/435,897 which is approximately .462. The probability that exactly one number will be chosen is 1/C(37,5) times [C(5,1) times C(32,4)] which is 179,800/435,897 which is approximately .4125. The probability of losing is the sum of these numbers which is approximately .8745. Of course this number is 1 - .1255, 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 using 37 numbers, we have the following:
ODDS VERSUS PROBABILITIES |
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| ODDS | PROBABILITY | DIFFERENCE | |
| Matching 5 | 0.000002296232239 | 0.000002296226967 | 0.000000000005273 |
| Matching 4 | 0.000367242007140 | 0.000367107189959 | 0.000000134817181 |
| Matching 3 | 0.011494253220056 | 0.011363636702299 | 0.000130616517757 |
| Matching 2 | 0.125000001047738 | 0.111111111938953 | 0.013888889108784 |
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 5/37 = .135135.... Since there have been 100 drawings at the time of this writing, x should have occurred 100 multiplied by .135135... times or 13.5 times. Compare this theoretical frequency with the actual frequencies given in the following table:
FREQUENCY OF OCCURRENCE OF |
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| 1 | 115 | 14 | 122 | 27 | 120 | ||
| 2 | 130 | 15 | 147 | 28 | 123 | ||
| 3 | 122 | 16 | 134 | 29 | 123 | ||
| 4 | 118 | 17 | 131 | 30 | 118 | ||
| 5 | 126 | 18 | 113 | 31 | 136 | ||
| 6 | 135 | 19 | 131 | 32 | 128 | ||
| 7 | 142 | 20 | 142 | 33 | 127 | ||
| 8 | 139 | 21 | 129 | 34 | 134 | ||
| 9 | 121 | 22 | 141 | 35 | 127 | ||
| 10 | 117 | 23 | 124 | 36 | 126 | ||
| 11 | 120 | 24 | 125 | 37 | 148 | ||
| 12 | 118 | 25 | 142 | ||||
| 13 | 133 | 26 | 120 | ||||
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 37 should be 19 and the standard deviation should be 10.82436. The actual average number chosen by the Texas Cash 5 Lottery machine for 100 drawings is ?? and the actual standard deviation is ??.
When the Chi-square test is run on the number frequencies for 100 drawings, we get the following results:
| X2 | = | (367 - 384.62)2/384.62 + (379 - 384.62)2/384.62 + |
| (402 - 384.62)2/384.62 + (355 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (378 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (390 - 384.62)2/384.62 + | ||
| (382 - 384.62)2/384.62 + (364 - 384.62)2/384.62 + | ||
| (391 - 384.62)2/384.62 + (387 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (355 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (378 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (390 - 384.62)2/384.62 + | ||
| (382 - 384.62)2/384.62 + (364 - 384.62)2/384.62 + | ||
| (391 - 384.62)2/384.62 + (387 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (355 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (378 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (390 - 384.62)2/384.62 + | ||
| (382 - 384.62)2/384.62 + (364 - 384.62)2/384.62 + | ||
| (391 - 384.62)2/384.62 + (387 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (355 - 384.62)2/384.62 + | ||
| (402 - 384.62)2/384.62 + (378 - 384.62)2/384.62 + | ||
| (401 - 384.62)2/384.62 | ||
| = | 7.66. |
According to a table of critical values of Chi square1, the Chi square value needs to be at least ?? to indicate non-randomness with a probability of at least .9, so it cannot be concluded at this point in time after 100 drawings that 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: If W players match 2 numbers, they each win $2.00 for a total of 2W dollars. Suppose A% of T is distributed among those who matched 5 numbers, B% of T is distributed among those who matched 4 numbers and C% of T is distributed among those who matched 3 numbers. If X people matched all 5 numbers, then each of them win .AT/X dollars. If Y people matched 4, then they each win .BT/Y. If Z people matched 3, then they win .CT/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/2Tand 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 using 37 numbers, the mathematical expectation may be seen in the following table:
| Probability | Prize | Product |
| X/2T | .AT/X | .AT/2T |
| Y/2T | .BT/Y | .BT/2T |
| Z/2T | .CT/Z | .CT/2T |
| 1/9 | $2.00 | .22 |
Since .AT + .BT + .CT = T - 2W, the sum if these expectations is (T - 2W)/2T + .22 = 1/2 - W/T. Thus, a player can expect to win less than 50 cents for every dollar spent on a ticket. How much less depends on how many players matched 2 numbers and won a guarenteed prize of $2.00
Conclusions:
It has been interesting to keep track of the number behavior for the new Texas CASH 5 Lottery using 37 numbers for 100 drawings. So far, there seems to be no indication of non-randomness in the number selection process and players can expect to win less than 50 cents on the dollar since the state takes 50% off the top before awarding prizes and pays $2.00 to those who match 2 numbers before dividing up the rest.
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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