Video Transcript
Suppose 𝐴 and 𝐵 are two mutually
exclusive events. Given the probability of 𝐴 union
𝐵 is equal to 0.93 and the probability of 𝐴 minus 𝐵 is equal to 0.39, find the
probability of 𝐵.
In this question, we are told that
the probability of 𝐴 or 𝐵, denoted 𝐴 union 𝐵, is equal to 0.93. We are also told that the
probability of 𝐴 occurring and 𝐵 not occurring, denoted the probability of 𝐴
minus 𝐵, is equal to 0.39. We are asked to find the
probability of 𝐵 occurring. As the two events 𝐴 and 𝐵 are
mutually exclusive, we know that the probability of 𝐴 union 𝐵 is equal to the
probability of 𝐴 plus the probability of 𝐵. It also means that the probability
of 𝐴 and 𝐵 occurring, denoted the probability of 𝐴 intersection 𝐵, is equal to
zero.
We can represent the probability of
these two events on a Venn diagram as shown. Since the events are mutually
exclusive, we can see from the diagram that the probability of 𝐴 minus 𝐵, that is,
the probability of 𝐴 occurring and 𝐵 not occurring, is equal to the probability of
𝐴. We know this is equal to 0.39. We also know that the the
probability of 𝐴 union 𝐵 is equal to 0.93. And since the events are mutually
exclusive, this equals the probability of 𝐴 plus the probability of 𝐵. We have 0.93 is equal to 0.39 plus
the probability of 𝐵. Subtracting 0.39 from both sides of
this equation gives us the probability of 𝐵 is equal to 0.93 minus 0.39, which is
equal to 0.54.
If 𝐴 and 𝐵 are mutually exclusive
events, the probability of 𝐴 union 𝐵 is equal to 0.93, and the probability of 𝐴
minus 𝐵 is 0.39, then the probability of 𝐵 is equal to 0.54.
Whilst it is not required in this
question, it is worth noting that the probability of neither 𝐴 nor 𝐵 occurring is
equal to 0.07. And this can be represented in the
completed Venn diagram as shown. The sum of all the probabilities in
the Venn diagram must equal one.
