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Chapter 3: Q. 3.14 (page 107)

Suppose that you are gambling against an infinitely rich adversary and at each stage you either win or lose 1 unit with respective probabilities p and 1 鈭� p. Show that the probability that you eventually go broke is 1 if p鈮�12and(qp)iif p>12where q = 1 鈭� p and i is your initial fortune.

Short Answer

Expert verified

This proves the statement.

Step by step solution

01

Given information

We know from Example 4jit's clear that

Pn,m=pPn-1,m+(1-p)Pn,m-1

where symbols have their usual meaning.

02

Step 2

Finally we obtain -

Pn,m=鈭�k=nm+n-1m+n-1kpk(1-p)m+n-1-k

Therefore we have, as the no. of trial tends to infinity, we get

P=1;p鈮�12qpi;p>12

where(m+n)鈫�鈭�

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