### Video Transcript

The probability that Ethan 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 letting M be the event that Ethan passes mathematics and P be the event that he passes physics. We are told in the question that the probability of him passing mathematics is 0.33. We are also told that the probability he fails physics is 0.32. This is known as the complement of π and is denoted π prime. The probability of π prime is therefore equal to 0.32.

Next, we recall that the complement formula states that the probability of π΄ prime is equal to one minus the probability of π΄. Rearranging this, the probability of π΄ is equal to one minus the probability of π΄ prime. As such, we can calculate the probability that Ethan passes physics by subtracting 0.32 from one. This is equal to 0.68.

We are also told in the question that the probability of Ethan passing at least one of the subjects is 0.71. This is known as the union of the two events and is denoted as shown. The probability of M union P is 0.71.

After clearing some space, letβs now consider what we need to calculate in this question. We are asked to find the probability that Ethan passes exactly one of the two subjects. We therefore need to find the probability that he passes mathematics only and the probability he passes physics only and then add these together.

To find the probability of each one of these, we can use the difference formula, which states the probability of π΄ minus π΅ is equal to the probability of π΄ minus the probability of π΄ intersection π΅, where the probability of π΄ minus π΅ is the probability that π΄ occurs and π΅ does not. And the probability of π΄ intersection π΅ is the probability that both events occur. It is this that we need to calculate next.

The addition formula of probability tells us that the probability of π΄ union π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ intersection π΅. This can be rearranged to find the probability of π΄ intersection π΅ as shown.

In this question, we can find the probability that Ethan passes both mathematics and physics by adding the probability he passes mathematics to the probability he passes physics and then subtracting the probability that he passes at least one of them. This is equal to 0.33 plus 0.68 minus 0.71, which is equal to 0.3. This is the probability that Ethan passes both subjects.

We are now in a position where we can find the probability of passing mathematics only and the probability of passing physics only. Firstly, the probability of passing mathematics and failing physics is equal to the probability of passing mathematics minus the probability of passing both. This is equal to 0.33 minus 0.3, which is equal to 0.03. We can repeat this process to find the probability of passing physics only. This is equal to 0.68 minus 0.3, which is equal to 0.38.

The probability of passing exactly one of the subjects is equal to the sum of these. Adding 0.03 and 0.38 gives us a final answer of 0.41. This is the probability that Ethan passes exactly one of the two subjects.

Whilst it is not required in this question, we could represent the information on a Venn diagram as shown, noting that we must add the probability that Ethan fails both exams, which is equal to one minus 0.71 or 0.29 outside of our two circles. We can then once again add 0.03 and 0.38 to find the probability that he passes exactly one of the two subjects.