2014 AMC 10B Problems/Problem 25
Problem
In a small pond there are eleven lily pads in a row labeled through
. A frog is sitting on pad
. When the frog is on pad
,
, it will jump to pad
with probability
and to pad
with probability
. Each jump is independent of the previous jumps. If the frog reaches pad
it will be eaten by a patiently waiting snake. If the frog reaches pad
it will exit the pond, never to return. what is the probability that the frog will escape being eaten by the snake?
Troll "Solution"
Using the techniques of a Markov chain, we can eventually arrive to the answer of
Actual (badly written) solution
Notice that the probabilities are symmetrical around the fifth lily pad. So if the frog is on the fifth lily pad, there is a chance that it escapes and a
that it gets eaten. Now, let
represent the probability that the frog escapes if it is currently on pad
. We get the following system of 5 equations:
We want to find
, since the frog starts at pad 1. Solving the above system yields
, so the answer is
.
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
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