# Question Video: Determining Probabilities using the Difference Rule and Other for Operations on Events Mathematics

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃(𝐴) = 0.39 and 𝑃(𝐵) is 0.82. Given that 𝑃(𝐴 ∩ 𝐵) = 0.23, what is the probability that only one of the events 𝐴 or 𝐵 occurs?

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### 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.