Suppose 𝐴 and 𝐵 are events in a
sample space which consists of equally likely outcomes. Given that the number of outcomes
of 𝐴 is three, the number of outcomes of 𝐵 is 11, the number of outcomes common to
both 𝐴 and 𝐵 is two, and the probability of 𝐴 union 𝐵 is three-eighths, find the
probability of 𝐴 bar union 𝐵.
We begin by recalling some of the
notation used in this question. 𝐴 bar, sometimes written 𝐴 prime,
denotes the complement of event 𝐴. This is the set of outcomes not in
event 𝐴. The union of events 𝐴 and 𝐵 is
the set of outcomes in event 𝐴 or in event 𝐵 or in both. One way of trying to answer a
question of this type is to begin by drawing a Venn diagram.
In this question, we are told there
are two events 𝐴 and 𝐵. There are three outcomes in event
𝐴, 11 outcomes in event 𝐵, and two outcomes that are common to both. The outcomes that are common to
both are found in the intersection of event 𝐴 and 𝐵. In our Venn diagram, this is found
in the overlap between the two circles.
Since there are three outcomes in
event 𝐴 and we know that two of these are common to event 𝐵, we can calculate the
number of outcomes just in event 𝐴 by subtracting two from three. This is equal to one. In the same way, since 11 minus two
is equal to nine, there are nine outcomes that occur in just event 𝐵.
We were told in the question that
the probability of 𝐴 union 𝐵 is equal to three-eighths. And from the Venn diagram, we see
that this corresponds to 12 outcomes. By considering equivalent fractions
and multiplying our numerator and denominator by four, we see that three over eight
is the same as 12 over 32. The probability of any event can be
found by dividing the number of successful outcomes by the total number of possible
outcomes. Since the numerator is equal to 12
and there are 12 outcomes in the union of events 𝐴 and 𝐵, we know there will be a
total of 32 outcomes. Subtracting 12 from 32 gives us
20. We can add this to our Venn
diagram. There are 20 outcomes that are not
in event 𝐴 or in event 𝐵.
We are trying to find the
probability of the union of the complement of 𝐴 and event 𝐵. The complement of event 𝐴 contains
29 outcomes. Therefore, the probability of this
is equal to 29 over 32. We want the union of this with
event 𝐵, noting that the probability of event 𝐵 is 11 over 32. Since nine of the outcomes occur in
both of these events, we do not need to count this twice. The number of outcomes that occur
in the union of these events is 20 plus nine plus two, which is equal to 31. The probability of the complement
of event 𝐴 union 𝐵 is therefore equal to 31 over 32.
An alternative way to answer this
question would be to use the addition rule of probability. This states that the probability of
𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the
probability of 𝐴 intersection 𝐵. This can be adapted for our
question such that the probability of 𝐴 bar union 𝐵 is equal to the probability of
𝐴 bar plus the probability of 𝐵 minus the probability of 𝐴 bar intersection
We have already worked out the
first two probabilities on the right-hand side of our equation. As mentioned earlier, there are
nine outcomes that are in the intersection of these two events. The probability of the complement
of 𝐴 union 𝐵 is therefore equal to 29 over 32 plus 11 over 32 minus nine over
32. As the denominators are the same,
we simply add and then subtract the numerators, giving us 31 over 32 once again.