Question: We can use the gambling interpretation given in Exercise 28 to find the expected number of tosses required to reach pattern B when we start

We can use the gambling interpretation given in Exercise 28 to find the expected number of tosses required to reach pattern B when we start with pattern A. To be a meaningful problem, we assume that pattern A does not have pattern B as a subpattern. Let EA(TB) be the expected time to reach pattern B starting with pattern A. We use our gambling scheme and assume that the first k coin tosses produced the pattern A. During this time, the gamblers made an amount AB. The total amount the gamblers will have made when the pattern B occurs is BB. Thus, the amount that the gamblers made after the pattern A has occurred is BB - AB. Again by the fair game argument, EA(TB) = BB-AB. For example, suppose that we start with pattern A = HT and are trying to et the pattern B = HTH. Then we saw in Exercise 28 that AB = 4 and BB = 10 so EA(TB) = BB-AB= 6. Verify that this gambling interpretation leads to the correct answer for all starting states in the examples that you worked in Exercise 28.

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