For two events 𝐴 and 𝐵, the
probability of 𝐴 dash, the complement of 𝐴, is equal to 0.4 and the probability of
𝐵 dash given 𝐴 dash is 0.6. Find the probability of 𝐴 dash
intersection 𝐵 dash.
The complement of 𝐴 means the
probability of 𝐴 not occurring. We recall that this is equal to one
minus the probability of 𝐴. The vertical line given that means
that we are dealing with conditional probability. The probability that 𝐵 does not
occur given that 𝐴 does not occur is 0.6. Another one of our probability
formulae states that the probability of 𝐴 given 𝐵 is equal to the probability of
𝐴 intersection 𝐵 divided by the probability of 𝐵. We can use a rearranged version of
that to help us work out the probability of 𝐴 dash intersection 𝐵 dash, the
probability of 𝐴 not happening and 𝐵 not happening.
Using our conditional probability
formula, we have that the probability of 𝐵 dash given 𝐴 dash is equal to the
probability of 𝐴 dash intersection 𝐵 dash divided by the probability of 𝐴
dash. Note that the complement of 𝐴 and
the complement of 𝐵 on the right-hand side are interchangeable as 𝐴 dash
intersection 𝐵 dash is the same as 𝐵 dash intersection 𝐴 dash. We can rearrange this formula by
multiplying both sides by the probability of the complement of 𝐴. Substituting in our values gives us
0.4 multiplied by 0.6. This is equal to 0.24. The probability of 𝐴 not occurring
and 𝐵 not occurring is 0.24 .