Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other Mathematics • 10th Grade

Video Transcript

The probability that event 𝐴 occurs is three-fifths. If event 𝐴 does not occur, then the probability of event 𝐵 occurring is two-thirds. What is the probability that event 𝐴 does not occur and event 𝐵 occurs?

We’re given two pieces of information in this question. The first is that the probability of event 𝐴 occurring is three-fifths. This is straightforward, and we can write this 𝑃 of 𝐴 is equal to three over five. The second piece of information is a conditional statement. This says that if event 𝐴 does not occur, then the probability of event 𝐵 occurring is two-thirds. And we write this as the probability of 𝐵 given not 𝐴 is equal to two-thirds. Remember, not 𝐴 is the complement of 𝐴 and this is written as 𝐴 with a bar on the top. And remember also that since the sum of all probabilities must equal one and that either 𝐴 or not 𝐴 must occur, then the probability of not 𝐴 is one minus the probability of 𝐴. And this will come in useful later on.

So we have the probability of 𝐴 is three-fifths and the probability of 𝐵 given not 𝐴 is two-thirds. The vertical bar in this expression reads given not, which is the conditional. We’re asked to find the probability of not 𝐴 and 𝐵. That’s the probability that event 𝐴 does not occur and event 𝐵 does occur. And this symbol that looks like an n represents “and” or the intersection of two events. So we have a simple probability, a conditional probability, and we’d like to find an intersection. And we can use the formula for conditional probability to find our intersection. This formula tells us that for two events 𝐶 and 𝐷, the probability of 𝐶 given that 𝐷 has occurred is the probability of 𝐶 and 𝐷 divided by the probability of 𝐷.

Now, if we let 𝐶 in the formula correspond to our event 𝐵 and 𝐷 in the formula is our event not 𝐴, then we can rewrite the formula as the probability of 𝐵 given not 𝐴 is equal to the probability of 𝐵 and not 𝐴 over the probability of not 𝐴. In our numerator, we know that the probability of 𝐵 intersection not 𝐴 is the same as the probability of not 𝐴 intersection 𝐵. So we can switch our 𝐵 and our not 𝐴 in the probability in the numerator.

And we see now that the numerator is actually what we’re looking for. That’s the probability of not 𝐴 intersection 𝐵. And we can make this the subject of our equation by multiplying through by the probability of not 𝐴. We can cancel this through on our right-hand side because the probability of not 𝐴 divided by the probability of not 𝐴 is equal to one so that we have the probability of not 𝐴 times the probability of 𝐵 given not 𝐴 is equal to the probability of not 𝐴 intersection 𝐵.

Now we know that the probability of 𝐵 given not 𝐴 is two-thirds. We also know that the probability of 𝐴 is three-fifths. And remembering that the probability of not 𝐴 is one minus the probability of 𝐴, we have the probability of not 𝐴 is one minus three-fifths, which is two-fifths. So in our formula, we have that the probability of not 𝐴 intersection 𝐵 is two-fifths times two-thirds. Multiplying fractions, we simply multiply the numerators and the denominators so that our numerator’s two times two is equal to four and our denominator is five times three, which is 15, so that the probability that event 𝐴 does not occur and event 𝐵 occurs is four out of 15.