Video Transcript
Suppose that 𝐴 and 𝐵 are events
with probabilities the probability of 𝐴 equals 0.39 and the probability of 𝐵 is
0.82. Given that the probability of 𝐴
intersection 𝐵 is 0.23, what is the probability that only one of the events 𝐴 or
𝐵 occurs?
To understand what is required in
this question, we will begin by drawing a Venn diagram. We are asked to work out the
probability that only one of the events 𝐴 or 𝐵 occurs. This means that either event 𝐴
occurs and 𝐵 does not, as shown by the orange-shaded section, or event 𝐵 occurs
and 𝐴 does not, as shown by the section shaded in pink. Each of these individually can be
represented by the difference of two events.
The difference rule of probability
states that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus
the probability of 𝐴 intersection 𝐵. Likewise, the probability of 𝐵
minus 𝐴 is equal to the probability of 𝐵 minus the probability of 𝐴 intersection
𝐵. In order to answer this question,
we need to find the sum of these two probabilities. The probability of 𝐴 minus 𝐵 is
equal to 0.39 minus 0.23. This is equal to 0.16. The probability of 𝐵 minus 𝐴 is
equal to 0.82 minus 0.23. This is equal to 0.59.
To calculate the probability that
only one of the events occurs, we need to add 0.16 and 0.59. This is equal to 0.75.
We can add the values calculated to
our Venn diagram. The probability that only event 𝐴
occurs is 0.16. The probability that only event 𝐵
occurs is 0.59. We are told in the question that
the probability of the intersection of these two events is 0.23. These three probabilities sum to
0.98. This means that the probability
that neither event 𝐴 nor event 𝐵 occurs is 0.02. We write this outside the two
circles in the Venn diagram.
To answer this question, the sum of
0.16 and 0.59 will give us the probability that only one of the events 𝐴 or 𝐵
occurs.
