2020 CIME II Problems/Problem 6
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
is simply
. For events
,
,
,
and probabilities
,
,
,
, the expected value when conducting the experiment is
. Thus, our expected value is just
Our answer is .
~mathboy100