2019 AIME II Problems/Problem 2
Problem 2
Lily pads lie in a row on a pond. A frog makes a sequence of jumps starting on pad . From any pad the frog jumps to either pad or pad chosen randomly with probability and independently of other jumps. The probability that the frog visits pad is , where and are relatively prime positive integers. Find .
Solution
Let be the probability the frog visits pad starting from pad . Then , , and for all integers . Working our way down, we find .
Solution 2(Casework)
Define a one jump to be a jump from k to K + 1 and a two jump to be a jump from k to k + 2.
Case 1: (6 one jumps) (1/2)^6 = 1/64
Case 2: (4 one jumps and 1 two jumps) 5C1 x (1/2)^5 = 5/32
Case 3: (2 one jumps and 2 two jumps) 4C2 x (1/2)^4 = 3/8
Case 4: (3 two jumps) (1/2)^3 = 1/8
Summing the probabilities gives us 43/64 so the answer is 107.
- pi_is_3.14
Solution 3 (easiest)
Let be the probability that the frog lands on lily pad . The probability that the frog never lands on pad is , so . This rearranges to , and we know that , so we can compute to be , meaning that our answer is
-Stormersyle
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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