2023 AMC 12A Problems/Problem 17
Contents
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
Initially, the probability of landing at and landing past (summing the infinte series) are exactly the same. Landing before 10 repeats this initial condition, with a different irrelevant scaling factor. Therefore, the probability must be .
Solution 2
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 (our expression is actually a general fact, not just limited to ). 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
~
Solution 3
In order to find the probability of landing on 10, we must multiply the amount of successful combinations by the probability of those combinations. Notice for any successful combination of steps, the probability must always be . Now, we only need to find the amount of possibilities for steps since we know the probability of each combination occurring is the same. This can be done using sticks and stones . Hence the final answer is or
~ShangJ2
Note that an alternative way of thinking the number of possible steps is that for each number from to , there are 2 possibilities: Flora lands on this number or Flora doesn't land on this number. Thus there are total possible combination of steps
~dwarf_marshmallow
Solution 4 (engineer's induction)
The probability frog lands on 1 is trivially
The probability frog lands on 2 is from the two cases 0-2 and 0-1-2.
The probability frog lands on 3 is from the cases 0-3, 0-1-3 and 0-2-3, 0-1-2-3.
The probability frog lands on 4 is from the cases 0-4, 0-1-4 and 0-3-4, 0-2-4, 0-1-2-4 and 0-1-3-4 and 0-2-3-4, 0-1-2-3-4.
It looks like the probability is regardless of the ending number. Therefore, we choose
~sirswagger21
Solution 5
No matter what the probability of getting to the end has a probability of
So the thing is how many ways to jump from the first spot to the last spot.
Given it requires steps to reach the end, there are ways to get the end.
~bluesoul
Solution 7 (Mathematician's Induction)
A more rigorous proof to solution 4.
Let represents the probability of Flora landing on number n
We hypothesize that
Base case: as shown in solution 4
Induction steps:
We assume that and is true (it's obvious here that this will add up to but to be 100% rigorous we do sum of geometric sequences B) Thus, since our base case it true, by induction every for and
~dwarf_marshmallow
Video Solution
https://youtu.be/dUyNe_2bBmM ~Education, the Study of Everything
Video Solution 1 by OmegaLearn
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by Solve It
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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