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Thread: Error in Mosteller book? Problem 37

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    Error in Mosteller book? Problem 37




    I have been reading this book since I was 13. Time to time I open it and solve one or two problems (if I can). Especially I like problem #37 about a guy who is in Vegas with $20 and needs $40 for a bus ticket. He decides to play roulette. The question is what is better to put all money on red (black) or play $1 at a time (Those times when they allowed $1 bet at the roulette are well behind us!). The result is correct it is about 11% chances if play cautiously. I was following his solution many times in the past, but just recently noticed that he arrives a the right number accidently. The formula he is using gives the correct result only if the amount is in the middle between $0 and the needed amount. In this case $40. So the symmetry brings the right answer but the formula itself is wrong in my opinion (and actually contradicts his own conclusions from problem #35 that it makes sense only if p > q).

    Anyone wants to discuss?

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    Re: Error in Mosteller book? Problem 37

    Actually did some more calculations and computer simulations to confirm, and found that Mosteller is correct. The formula works both ways - when p > q, and when q > p. However, I still don't understand why. Based on the problem 35, one cannot use this logic...

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    Re: Error in Mosteller book? Problem 37

    I do not have the book. It will help to discuss the problem if you can type out the solution with reference to the one in #35. It should be related to the ruin problem in Markov Chain.

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    Re: Error in Mosteller book? Problem 37

    Quote Originally Posted by BGM View Post
    I do not have the book. It will help to discuss the problem if you can type out the solution with reference to the one in #35. It should be related to the ruin problem in Markov Chain.
    Sorry, I am not a professional. Don't know what Markov Chain is. I just love probabilities. I could have scanned a couple of pages from the book, but I have a very old 1975 edition in Russian.

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    Re: Error in Mosteller book? Problem 37


    I finally figured this issue. Mosteller is correct and the formula works. However he is missing one transitional step from problems 35 and 36 to problem 37. When he is inputting 'p' and 'q' into the formula from problem 36, it is not correct. There must be another formula derived, which is almost identical, except for 'm' and 'n'. When 'q' (player's probability to win at any given step) is smaller than 'p' (casino's probability), then 'm' from problem 36 becomes actually 'n'. This is due to two facts: one is the symmetry of the problem, and two - the fact that 'q' must be always smaller than 'p'. Otherwise, based on problem 35, solution does not make sense. So my initial suspicion was right - there was something missing. You could not just use formula from 36 and input there 'q' and 'p' where 'q' > 'p', like he did. The correct result in such case is only possible for the m == n (in his case both = 20). As soon as you use other (non symmetrical) conditions, for example, a player has $15 rather than $20, and still needs to get to $40, it becomes evident.

    For those, who are interested and have a book, what Mosteller should have done before moving to problem 37, was to have solved problem 36 in reverse - with the same conditions, except player M has a chance to win at his move is 1/3 (originally 2/3), and player N has a chance of 2/3 (originally 1/3). This would be a perfect prototype for problem 37, and would give the correct formula.

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