7 Keep tossing a fair coin till you get at least one H and a

7. Keep tossing a fair coin till you get at least one H and at least one T. What is the expected number of tosses?

7. Keep tossing a fair coin till you get at least one H and at least one T. What is the expected number of tosses?

Solution

A difference equation is often useful here. Let a = expected number of throws to first head. We must make 1 throw at least and we have probability 1/2 of a head and probability 1/2 of returning to a, so a = (1/2)1 + (1/2)(1 + a) (1/2)a = 1 a = 2. Let E = expected number of throws to 2 consecutive heads. Consider that we have just thrown a head and what happens on the next throw. We are dealing with the (a + 1)th throw, with probability 1/2 this is not a head and we return to E. So E = (1/2)(a + 1) + (1/2)(a + 1 + E) (1/2)E = a + 1 E = 2(a + 1) and now putting in the value a = 2 we get E = 2(3) = 6 Expected throws to 2 consecutive heads is 6.