smc

Question

Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process ab, where a and b are relatively prime positive integers. What is a + b? (For example, he succeeds if his sequence of tosses is HTHHHHHH.)

Accepted Answer

For 6 to 8 heads, we are guaranteed to hit 4, so the sum here is 82+81+80=28+8+1=37. For 4 heads, you have to hit the 4 heads at the start so there's only one way, 1. For 5 heads, we either start off with 4 heads, which gives us _4C_1=4 ways to arrange the other flips, or we start off with five heads and one tail, which has 6 ways minus the 2 overlapping cases, HHHHHTTT and HHHHTHTT. Total ways: 8. Then we sum to get 46. There are a total of 2^8=256 possible sequences of 8 coin flips, so the probability is 46256=23128. Summing, we get 23+128=151.