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 $10?$

$\dfrac{5}{512}$

$\dfrac{45}{1024}$

$\dfrac{127}{1024}$

$\dfrac{511}{1024}$

$\dfrac{1}{2}$

Answer: E

Concepts:recursive probability recursion

Difficulty rating: 1910

Solution:

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

We prove by induction that $a_n=\tfrac12$ for every $n\ge 1.$ The case $n=1$ is immediate. If the claim holds through $n-1,$ then $\begin{aligned} a_n&=\frac{1}{2^n}a_0 +\sum_{k=1}^{n-1}\frac{1}{2^k}\cdot\frac12\\ &=\frac{1}{2^n} +\frac12\left(1-\frac{1}{2^{n-1}}\right)\\ &=\frac12. \end{aligned}$

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

Thus, the correct answer is E.

2023 AMC 12A Problem 17

Answer: E

Difficulty rating: 1910

Solution:

Problem 17 in Other Years