### Video Transcript

The probability that James passes
mathematics is 0.33, and the probability that he fails physics is 0.32. Given that the probability of him
passing at least one of them is 0.71, find the probability that he passes exactly
one of the two subjects.

We will begin by defining event π΄
as passing mathematics and event π΅ as passing physics. It is important to note that we are
only considering two possible outcomes, whether the student passes or fails it. We arenβt interested in the grade
or score, just whether they pass or fail. This means that the probability of
event π΄, James passing mathematics, is 0.33. We are given the probability that
James fails physics. This is equal to 0.32. This is known as the complement of
event π΅ and is written the probability of π΅ bar. This is equal to 0.32.

We know that the probability of the
complement is equal to one minus the probability of the event. In this case, 0.32 is equal to one
minus the probability of π΅. Rearranging this equation, we have
the probability of event π΅ is equal to one minus 0.32 which in turn is equal to
0.68. The probability of James passing
physics is 0.68.

We are also told that the
probability of James passing at least one subject is 0.71. This is the same as saying that
James passes either maths or physics or both. This can be written as the
probability of π΄ union π΅. We can then use the addition rule
of probability to calculate the probability of π΄ intersection π΅. This is equal to the probability of
π΄ plus the probability of π΅ minus the probability of π΄ union π΅. Substituting in the values we know,
this is equal to 0.33 plus 0.68 minus 0.71 which is equal to 0.3 or 0.30. The probability of π΄ intersection
π΅ is 0.3.

At this stage, we can proceed in
one of two ways. Firstly, we can use the difference
rule for probability. This states that the probability of
π΄ minus π΅ is equal to the probability of π΄ minus the probability of π΄
intersection π΅. We also have the probability of π΅
minus π΄ is equal to the probability of π΅ minus the probability of π΄ intersection
π΅. Finding the sum of these two events
will give us the probability that James passes exactly one of the two subjects. Letβs begin by calculating the
probability that he only passes mathematics. The probability of π΄ minus π΅ is
equal to 0.33 minus 0.3. This is equal to 0.03. Now letβs consider the probability
that James passes only physics. This is equal to 0.68 minus 0.3,
which is equal to 0.38.

By finding the sum of these two
values, we can conclude that the probability James passes exactly one of the two
subjects is 0.41. We can demonstrate this solution on
a Venn diagram. We know that the probability of
events π΄ and π΅ both occurring is 0.3, the probability of only event π΄ occurring,
James only passing maths, is 0.03, and the probability of only event π΅ occurring is
0.38. The three values on the Venn
diagram sum to 0.71, the probability of π΄ or π΅ occurring. As we know that all probabilities
sum to one, the probability that neither event π΄ nor event π΅ occurs is 0.29. This is the probability that James
fails both subjects. Adding 0.03 and 0.38, this confirms
that the probability James passes exactly one of his two subjects is 0.41.