2019 AMC 12B Problems/Problem 16
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[hide]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 1
First, notice that Fiona, if she jumps over the predator on pad , 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 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 ), or she can go (probability of ). 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 go (probability of ). Any other option results in her death to a predator.
Thus, the final answer is
Solution 2
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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