2020 CIME II Problems/Problem 6
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Problem
An infinite number of buckets, labeled ,
,
,
, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket
is
. Given that all three balls land in the same bucket
and that
is even, then the expected value of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Solution
The probability that all three balls land in box is
. This means that the probability that the three balls land in the same even box is
. This means that the probability that all three balls land in box
\emph{given that they land in the same even box}