If #A# and #B# are two events such that #P(A)ne0# and #P(B|A)=1#, then?

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1 Answer
Steve M
Sep 27, 2017

#B subset A#

Explanation:

Using the conditional probability formula:

# P( X | Y ) = ( P (X nn Y) ) / ( P(Y) ) #

We have:

# P( B | A ) = 1 => ( P (B nn A) ) / ( P(A) ) = 1#

And therefore:

# P (B nn A) = P(A) #

If we now think about the underlying set theory then clearly:

# {B nn A} = {A} #

As #{B nn A}# must be larger set or the same as #{A}#, so then #{A nn B}# must be comprised only of elements of #{A}# alone, i.e., #{B}# is a subset of #{A}#, Thus:

#B subset A#

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