math319_S10_Worksheet6

# math319_S10_Worksheet6 - = if q p 6 = 1 if q p = 1 4 Let Q...

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Probability and Statistics – Math319, Spring 2010 Worksheet 6 April 1, 2010 Gambler’s Ruin We consider now a more general version of the Gambler’s Ruin problem where each coin ﬂip results in heads with probability p . 1. Then P i = , and, as before, P i +1 = and P i - 1 = . So, in terms of P i +1 and P i - 1 , P i = , which can be rearranged to give P i +1 - P i = . 2. Summing successive versions of this last formula, and taking advantage of cancelations, P 2 - P 1 = P 3 - P 2 = . . . = gives P i - P 1 = = and so P i = = , if q p 6 = 1 . So, altogether we have P i = if q p 6 = 1 if q p = 1 .

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3. In particular, P N = if q p 6 = 1 if q p = 1 , which implies that P 1 = if q p 6 = 1 if q p = 1 , and so P i
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Unformatted text preview: = if q p 6 = 1 if q p = 1 . 4. Let Q i be the probability that C ends up with all the money when G starts with i and C with N-i units. Then, by symmetry with the above situation, Q i = if p q 6 = 1 if p q = 1 . 5. So, if p q 6 = 1 , P i + Q i = , = , = , = . 6. Also, when p q = 1 , P i + Q i = , = . 7. So, we see that, with probability , either G or C will end up with all the money. I.e. there is probability of the game continuing indeﬁnitely....
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## This note was uploaded on 10/14/2010 for the course MATH 319 taught by Professor Clionagolden during the Spring '10 term at Bard College.

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math319_S10_Worksheet6 - = if q p 6 = 1 if q p = 1 4 Let Q...

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