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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 − p. Show that the probability that you eventually go broke is 1 if p≤12and(qp)iif p>12where q = 1 − 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=∑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≤12qpi;p>12

where(m+n)→∞

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Most popular questions from this chapter

With probability .6, the present was hidden by mom; with probability .4, it was hidden by dad. When mom hides the present, she hides it upstairs 70percent of the time and downstairs 30percent of the time. Dad is equally likely to hide it upstairs or downstairs.

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