Difference between revisions of "2019 AIME II Problems/Problem 2"
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+ | ==Solution 3 (easiest)== | ||
+ | Let <math>P_n</math> be the probability that the frog lands on lily pad <math>n</math>. The probability that the frog never lands on pad <math>n</math> is <math>\frac{1}{2}P_{n-1}</math>, so <math>1-P_n=\frac{1}{2}P_{n-1}</math>. This rearranges to <math>P_n=1-\frac{1}{2}P_{n-1}</math>, and we know that <math>P_1=1</math>, so we can compute <math>P_7</math> to be <math>\frac{43}{64}</math>, meaning that our answer is <math>\boxed{107}</math> | ||
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+ | -Stormersyle | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=II|num-b=1|num-a=3}} | {{AIME box|year=2019|n=II|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:02, 22 March 2019
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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