Question

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\dfrac{1}{2^m}.$ What is the probability that Flora will eventually land at $10?$

$\dfrac{5}{512}$

$\dfrac{45}{1024}$

$\dfrac{127}{1024}$

$\dfrac{511}{1024}$

$\dfrac{1}{2}$

Solution:

Let $a_n$ be the probability that Flora ever lands exactly on $n,$ with $a_0=1.$ Conditioning on the first jump, $a_n=\sum_{k=1}^{n}\dfrac{1}{2^k}\,a_{n-k}.$

Then $a_1=\tfrac12,$ $a_2=\tfrac12,$ and by induction $a_n=\tfrac12$ for all $n\ge 1:$ each new term averages the previous values, all equal to $\tfrac12.$

Hence the probability of landing on $10$ is $\dfrac12.$

Thus, the correct answer is E.

2023 AMC 12A Problem 17

Solution:

Problem 17 in Other Years