25. A coin is tossed repeatedly, heads appearing on each toss with probability p. A gambler starts with initial fortune k (where 0 ?

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25. A coin is tossed repeatedly, heads appearing on each toss with probability p. A gambler starts
with initial fortune k (where 0 <k < N); he wins one point for each head and loses one point for
each tail. If his fortune is ever 0 he is bankrupted, whilst if it ever reaches N he stops gambling to buy
a Jaguar. Suppose that p < . Show that the gambler can increase his chance of winning by doubling
the stakes. You may assume that k and N are even.
What is the corresponding strategy if p > ?
Transcribed Image Text:25. A coin is tossed repeatedly, heads appearing on each toss with probability p. A gambler starts with initial fortune k (where 0 <k < N); he wins one point for each head and loses one point for each tail. If his fortune is ever 0 he is bankrupted, whilst if it ever reaches N he stops gambling to buy a Jaguar. Suppose that p < . Show that the gambler can increase his chance of winning by doubling the stakes. You may assume that k and N are even. What is the corresponding strategy if p > ?
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