1983 AHSME Problems/Problem 26
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Problem
The probability that event occurs is
; the probability that event B occurs is
.
Let
be the probability that both
and
occur. The smallest interval necessarily containing
is the interval
Solution
Firstly note that and
, as clearly the probability that both
and
occur cannot be more than the probability that
or
alone occurs. The more restrictive condition is
, since
.
Now, by the Inclusion-Exclusion Principle, we also have and as a probability must be non-negative,
, so
. Therefore, combining our inequalities gives
, which is answer choice
.