2023 AMC 12A Problems/Problem 17
Problem
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 with probability
.
What is the probability that Flora will eventually land at 10?
Solution 1
Let's denote as the probability of reaching
from
. We immediately see that
, and
, since there's only one way to get to 1 from 0. Just jump!
Now, let's write an expression for . Suppose we know
.
The probability of reaching 10 from some integer will be
(use the formula given in the problem!)
The probability of reaching that integer
from
is going to be
. Then, the probability of going from
will be
We want the probability of reaching 10 from anywhere though, so what we can do is sum over all passing points
, i.e.
Now that we have expressed our problem formally, we can actually start solving it!
Let's calculate .
Hmm, we see that the first 8 terms of
are exactly the first 8 terms of
. Let's substitute it in.
Isn't that interesting. Turns out, this reasoning can be extended all the way to
.
It breaks at , since
. Anyway, with this, we see that the answer is just
~
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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