2016 AMC 12A Problems/Problem 19
Problem
Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves unit in the positive direction; when he gets tails, he moves unit in the negative direction. The probability that he reaches at some time during this process where and are relatively prime positive integers. What is (For example, he succeeds if his sequence of tosses is )
Solution
For to heads, we are guaranteed to hit heads, so the sum here is .
For heads, you have to hit the heads at the start so there's only one way, .
For heads, we either start off with heads, which gives us ways to arrange the other flips, or we start off with five heads and one tail, which has ways minus the overlapping cases, and . Total ways: .
Then we sum to get . There are a total of possible sequences of coin flips, so the probability is . Summing, we get .
See Also
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.