If 𝐴 and 𝐵 are two events of a sample space 𝑆 of a random experiment where 𝐴 is a subset of 𝐵, then the probability of 𝐵 given 𝐴 equals blank. Is it a) the probability of 𝐴, b) the probability of 𝐵, c) the probability of 𝐴 minus 𝐵, or d) the probability of 𝑆?
Let’s sketch what we know. If 𝑆 is our sample space, we have event 𝐴 which is a subset of event 𝐵. We’re looking for the probability of 𝐵 happening, given that 𝐴 has happened. We know the probability of 𝐵 given 𝐴 equals the probability of the intersection of 𝐴 and 𝐵 divided by the probability of 𝐴. What is the intersection of 𝐴 and 𝐵 in our problem?
All of 𝐴 is the intersection. If the intersection of 𝐴 and 𝐵 is the probability of 𝐴 and we divide that by the probability of 𝐴, we get one, a probability of one. We don’t see the number one as one of the answer choices. However, one of these probabilities does equal one. We were told that our sample space is 𝑆. The probability of the sample space equals one. And that means the probability of 𝐵 given 𝐴 is equal to the probability of the whole sample space. At first, this might not seem intuitive to us. So let’s consider an example.
Let’s say our experiment is a fair die roll. And 𝐴 is rolling a two and 𝐵 is rolling an even number. In this case, 𝐴 is a subset of 𝐵 because two is an even number. What is the probability of 𝐵 given 𝐴 for this scenario? Well, it’s the probability that you rolled an even value if you know you rolled a two. And that probability is one. Since 𝐴 is a subset of 𝐵 and you know you have 𝐴, you also have 𝐵.
The probability of 𝐵 given 𝐴 equals one, the probability of the sample space.