2019 AIME I Problems/Problem 5
Contents
Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point
, it moves at random to one of the points
,
, or
, each with probability
, independently of its previous moves. The probability that it will hit the coordinate axes at
is
, where
and
are positive integers. Find
.
Solution
One could recursively compute the probabilities of reaching as the first axes point from any point
as
for
and the base cases are
for any
not equal to one.
We then recursively find
so the answer is
.
If this algebra seems intimidating, you can watch a nice pictorial explanation of this by On The Spot Stem. https://www.youtube.com/watch?v=XBRuy3_TM9w
Solution 2
Obviously, the only way to reach (0,0) is to get to (1,1) and then have a chance to get to (0,0). Let x denote a move left 1 unit, y denote a move down 1 unit, and z denote a move left and down one unit each. The possible cases for these moves are
and
. This gives a probability of
to get to
. The probability of reaching
is
. This gives
.
- The MAA should have specified that
is not divisible by
or the greatest common divisor of
and
was 1.
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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