Video Transcript
Suppose 𝐴 and 𝐵 are two
events. Given that 𝐴 intersection 𝐵 is
the empty set, the probability of 𝐴 prime is 0.66, and the probability of 𝐵 prime
is 0.79, find the probability of 𝐵 minus 𝐴.
Before trying to answer this
question, let’s recall some of the notation. We know that 𝐴 prime and 𝐵 prime
are the complement of events 𝐴 and 𝐵, respectively. And we also know that the
probability of the complement of event 𝐴 is equal to one minus the probability of
event 𝐴. Using the information given, we can
therefore calculate the probability of event 𝐴 along with the probability of event
𝐵.
Firstly, we have 0.66 is equal to
one minus the probability of 𝐴. Rearranging this equation, we have
the probability of 𝐴 is equal to one minus 0.66. This is equal to 0.34. In the same way, 0.79 is equal to
one minus the probability of event 𝐵. The probability of 𝐵 is therefore
equal to one minus 0.79, which is equal to 0.21.
We are also told that 𝐴
intersection 𝐵 is equal to the empty set. This means that there are no
elements in event 𝐴 and event 𝐵. And we can therefore say that the
two events are mutually exclusive. And the probability of 𝐴
intersection 𝐵 is therefore equal to zero. When representing this on a Venn
diagram, there is no overlap as shown. We can fill in the fact that the
probability of event 𝐴 is 0.34 and the probability of event 𝐵 is 0.21. We can complete the Venn diagram by
filling in the probability that neither event 𝐴 nor event 𝐵 occur. This is equal to 0.45.
We are asked to find the
probability of 𝐵 minus 𝐴. And using the difference formula,
we know this is equal to the probability of 𝐵 minus the probability of 𝐴
intersection 𝐵. Substituting in the values we know,
this is equal to 0.21 minus zero, which is just equal to 0.21. This leads us onto an important
rule. If two events 𝐴 and 𝐵 are
mutually exclusive, then the probability of 𝐵 minus 𝐴 is simply equal to the
probability of 𝐵. Likewise, the probability of 𝐴
minus 𝐵 is equal to the probability of 𝐴.
