### Video Transcript

A and B are independent events, where the probability of A is equal to a half and the probability of B is equal to three-sevenths. What is the probability that event A occurs but event B does not?

So first of all, we need to look at what independent events are. And independent events are events where the outcome of one event does not affect the outcome of the other. So now, in this question, what we’re taking a look at is we want to find the probability that event A occurs. But we want to find the complement of the probability that event B occurs. So we want to find the probability that event B does not occur.

So therefore, as I said, we know that the probability of A is equal to a half and the probability of not B is gonna be equal to one or seven-sevenths minus three-sevenths. And we get that because we know that the probability of A is equal to one minus the probability of not A or the complement of A. So therefore, we can say that the complement of B is equal to four-sevenths. Okay, great. So we now have both of our probabilities that we’re looking for.

So what we’re looking to find is the probability that A occurs and the probability of not B or the complement of B occurs. And the key here is the word “and” because we have something called an “and” rule when we’re looking at probability. And this “and” rule applies when we’re looking at independent events. And what this “and” rule states is that the probability of A and B is equal to the probability of A multiplied by the probability of B.

So therefore, in our case, the probability of A and the complement of B is gonna be equal to these two things and multiplied together. And we have the values for these because we worked out the probability of not B or the complement of B. So it’s gonna be equal to a half multiplied by four-sevenths. Well, if we multiply the numerators and the denominators, this’s gonna give us four over 14. And this will cancel down to two-sevenths. And we could’ve got that originally because if we think of a half multiplied by four-sevenths, well, it’s a half of four-sevenths, which is gonna be two-sevenths.

So therefore, we can say that the probability that event A occurs but event B does not is two over seven or two-sevenths.