Sanderson M. Smith

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QUESTION FROM PARENT INVOLVING SUPERLOTTO PLUSWhat appears below (in the box) is part of a note I received from the father of one of my students.

Now, I have a question for you - which I hope you will be able to answer in some manner - based on the fact that you have written articles and done probability studies on the California Lottery (per your web site). We have a California Lottery outlet on one of our properties. It is actually on the California/Nevada state line just outside of Reno, and it is either the second or third largest lottery outlet (in terms of tickets sold) in

California. It has been open since December 1998, and we have sold approximately 7,588,000 SuperLotto tickets during that period. However, we have never sold a winning ticket, and to the best of knowledge, we have only sold a small handful of winning tickets above the $600 threshold, with the largest being $1400. The Lottery has told us we are long overdue for a winner, and they are somewhat surprised that, based on our volume, we haven't had one yet. Is it possible for you to calculate in some type of meaningful mathematical terms what the probability is for someone to hit a winning at that site and when it could happen (other than at any time)? If this is too complex, don't bother. I just thought that based on your recent analysis of roulette and your Lottery work you might have some ideas.What follows represents my response.

Hi (name removed):

First off, I hope it is OK if I send your question (and my response) to my STATISTICS students. I will also send it to others on my ListServes since it highlights a number of fascinating mathematical points, including what is know as the gambler's fallacy.

You question is an interesting one, and puzzles me just a bit.

The puzzling thing is the statement of the Lottery Commission. If, based on 7,588,000 tickets sold, they say your site (outlet) is overdue for a winner, this is a statistically inaccurate statement. Since the whole process is random, your site is no more "overdo" than any other site that sells tickets. This is similar to the gambler's fallacy applied to games, such as roulette (which we have played in Algebra II and in Statistics, and we have examined the mathematics of the game.) A famous gambler's fallacy theory in roulette is to watch the games until RED comes up on three consecutive rolls of the ball, then bet on BLACK using the (totally false) reasoning that BLACK is more likely since the probability of getting 4 REDS in a row is small. So, your site is no more likely to win than any other site that sells roughly the same number of tickets, even if the other site has had multiple winners.

Now, California SuperLotto has you choose five numbers from the set

S = {1, 2, 3,..., 45, 46, 47}

and one number (the MEGA number) from the set

M = {1,2,3, ..., 25, 26, 27}

If your set of 6 numbers (five plus the MEGA) match a similar set chosen randomly by the State, you win millions of dollars.

The number of possible number sets is

(

_{47}C_{5})(_{27}C_{1}) = (1,533,939)(27) = 41,416,353Hence, the probability that a ticket does not win is 41416352/41416353 = 0.9999999759. The probability that one would not win if one purchased 7,588,000 tickets is

(0.9999999759)

^{7,588,000 }= 0.83259, or approximately 83%In other words, with the purchase of 7,588,000 tickets, the probability that you would have a winner is 100% - 83% = 17%. Hence, it is not surprising that your outlet has not produced a winner yet.

When will your site produce a winner? There is no mathematical way to determine that. It could be tomorrow, a year from now, ten years from now, never, etc., etc.

I hope this is helpful to you.

Comments on the letter and my response are welcome.

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