Question Video: Using the Addition Rule to Determine the Probability of Exactly One of Two Events Occurring Mathematics

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 𝐴 as passing mathematics and event 𝐵 as passing physics. It is important to note that we are only considering two possible outcomes, whether the student passes or fails it. We aren’t interested in the grade or score, just whether they pass or fail. This means that the probability of event 𝐴, 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 𝐵 and is written the probability of 𝐵 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 𝐵. Rearranging this equation, we have the probability of event 𝐵 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 𝐴 union 𝐵. We can then use the addition rule of probability to calculate the probability of 𝐴 intersection 𝐵. This is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 union 𝐵. 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 𝐴 intersection 𝐵 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 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. We also have the probability of 𝐵 minus 𝐴 is equal to the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. Finding the sum of these two events will give us the probability that James passes exactly one of the two subjects. Let’s begin by calculating the probability that he only passes mathematics. The probability of 𝐴 minus 𝐵 is equal to 0.33 minus 0.3. This is equal to 0.03. Now let’s 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 𝐴 and 𝐵 both occurring is 0.3, the probability of only event 𝐴 occurring, James only passing maths, is 0.03, and the probability of only event 𝐵 occurring is 0.38. The three values on the Venn diagram sum to 0.71, the probability of 𝐴 or 𝐵 occurring. As we know that all probabilities sum to one, the probability that neither event 𝐴 nor event 𝐵 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.