2019 AMC 12B Problems/Problem 16

Problem

Lily pads numbered from to lie in a row on a pond. Fiona the frog sits on pad , a morsel of food sits on pad , and predators sit on pads and . At each unit of time the frog jumps either to the next higher numbered pad or to the pad after that, each with probability , independently from previous jumps. What is the probability that Fiona skips over pads and and lands on pad ?

Solution

First, notice that Fiona, if she jumps over the predator on pad , \textbf{must} land on pad . Similarly, she must land on if she makes it past . Thus, we can split it into smaller problems counting the probability Fiona skips , Fiona skips (starting at ) and \textit{doesn't} skip (starting at ). Incidentally, the last one is equivalent to the first one minus .

Let's call the larger jump a -jump, and the smaller a -jump.

For the first mini-problem, let's see our options. Fiona can either go (probability of \frac{1}{8}), or she can go (probability of \frac{1}{4}). These are the only two options, so they together make the answer . We now also know the answer to the last mini-problem ().

For the second mini-problem, Fiona \textit{must} go (probability of \frac{1}{4}). Any other option results in her death to a predator.

Thus, the final answer is