Suppose 𝑥 and 𝑦 are two events with probabilities 𝑃 of 𝑥 equals 0.49 and 𝑃 of 𝑦 equals 0.48. Given that 𝑃 of 𝑥 ∪ 𝑦 is 0.95, determine 𝑃 of the complement of 𝑥 ∩ 𝑦.
We’ve been given some information about two events 𝑥 and 𝑦. We know that the probability that 𝑥 occurs is 0.49 and the probability that 𝑦 occurs is 0.48. The probability of 𝑥 ∪ 𝑦 is 0.95. Now if we think about this in terms of a Venn diagram, 𝑥 ∪ 𝑦 is the shaded region. It’s everything in 𝑥, everything in 𝑦, and everything in the overlap. Now, we’re looking to find the probability of the complement of 𝑥 ∩ 𝑦, in other words, the probability that 𝑥 ∩ 𝑦 does not occur. And so it follows that we should begin by finding the probability of 𝑥 ∩ 𝑦. And in Venn diagram terms, that’s the overlap of our two circles.
Now, the general probability addition rule for the union of two events says that the probability of 𝑥 ∪ 𝑦 is equal to the probability of 𝑥 plus the probability of 𝑦 minus the probability of 𝑥 ∩ 𝑦. Now, it’s important to realize that this is the probability of two nonmutually exclusive events. So we really should check whether our events are mutually exclusive or not. Now, if the sum of the probabilities is equal to the probability of the union of these two events, then they are mutually exclusive. Now, in this case, 0.49 add 0.48 is not 0.95. And we can use our addition formula.
We substitute in and we see that 0.95 is equal to 0.49 plus 0.48 minus the probability of 𝑥 ∩ 𝑦. We simplify a little on the right-hand side becomes 0.97 minus the probability of 𝑥 ∩ 𝑦. We’ll subtract 0.95 from both sides and add the probability of 𝑥 ∩ 𝑦. And we see that the probability of 𝑥 ∩ 𝑦 is 0.97 minus 0.95, which is 0.02. And we found the probability of 𝑥 ∩ 𝑦; it’s 0.02. The probability that both events occur at the same time is 0.02.
Now, of course, the question’s actually asking us to find the probability of the complement of 𝑥 ∩ 𝑦, the probability that 𝑥 ∩ 𝑦 does not happen. Well, we know we can find this by subtracting the probability of 𝑥 ∩ 𝑦 from one. That gives us 0.98. And so we’ve answered the question. The probability of the complement of 𝑥 ∩ 𝑦 is 0.98.