Question Video: Calculating Dependent Probabilities Mathematics

Video Transcript

For two events 𝐴 and 𝐵, the probability of 𝐴 equals 0.6, the probability of 𝐵 equals 0.5, and the probability of 𝐴 union 𝐵 equals 0.7. Work out the probability of 𝐴 given 𝐵.

In this question, we’re given that there are two events 𝐴 and 𝐵. We’re also given the probability of these two different events happening. And we’re then asked to work out a conditional probability, the probability of 𝐴 given 𝐵. We should remember that there’s a formula to help us calculate conditional probability. For the probability of 𝐴 given 𝐵, we calculate the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. At this point, we may notice that we have a bit of a problem. We need the probability of 𝐴 intersection 𝐵, but instead we’re given the probability of 𝐴 union 𝐵.

Let’s see if we can model this scenario using a Venn diagram. The probability of event 𝐴 is 0.6, and the probability of event 𝐵 is 0.5. If we were just to add the probability of 𝐴 and the probability of 𝐵, we’d have a problem. Everything in this central section, which represents the probability of 𝐴 intersection 𝐵 or the probability of 𝐴 and 𝐵, would be counted twice. And this leads us to a helpful formula which connects the probability of 𝐴 intersection 𝐵 and the probability of 𝐴 union 𝐵. On the right-hand side, if we add together the probability of 𝐴 and the probability of 𝐵 and then subtract the probability of 𝐴 intersection 𝐵, then what we have is the probability of 𝐴 union 𝐵, and that’s on the left-hand side.

As we’re given the probabilities for 𝐴, 𝐵, and 𝐴 union 𝐵, then we can fill these into the formula to find the value of the probability of 𝐴 intersection 𝐵. 0.7 equals 0.6 plus 0.5 subtract the probability of 𝐴 intersection 𝐵. Then, we can simplify 0.6 plus 0.5 to 1.1. We can add the probability of 𝐴 intersection 𝐵 and subtract 0.7 from both sides of the equation to give us that the probability of 𝐴 intersection 𝐵 is equal to 0.4. Now that we know this value, we can plug it into the formula to find the probability of 𝐴 given 𝐵. So the probability of 𝐴 given 𝐵 is equal to 0.4, since that’s the probability of 𝐴 intersection 𝐵, divided by 0.5, since that’s the probability of 𝐵.

Remember that the probability of 𝐴 given 𝐵 really means that given we’ve got an outcome that’s part of event 𝐵, then what’s the probability that it’s also part of event 𝐴? So we have the proportion of 0.4 over 0.5. Of course, we can write our fraction 0.4 over 0.5 in a nicer way by multiplying the numerator and denominator by 10. So the answer is that the probability of 𝐴 given 𝐵 is equal to four-fifths.