Suppose 𝐴 and 𝐵 are two events. Given that 𝐵 is a subset of 𝐴, the probability of 𝐵 occurring is four-ninths, and the probability of 𝐴 minus 𝐵 is one-fifth, determine the probability of 𝐴.
Let’s go back to this first bit of information. This shape that looks a little like the letter C means a subset of. Now, in terms of probability, we can say that this means that the probability of 𝐵 occurring has to be less than or equal to the probability of 𝐴 occurring. But we can also think about what this might look like in Venn diagram form.
For 𝐵 to be a subset of 𝐴, the Venn diagram might look a little bit like this. We’re told the probability that 𝐵 occurs is four-ninths and that the probability of the difference of 𝐴 and 𝐵 is one-fifth. So, we can say that a fifth must go here. This means that the probability of 𝐴 occurring is the sum of these two. It’s a fifth plus four-ninths. We create a common denominator of 45. We multiply the numerator and denominator of our first fraction by nine and of our second by five. We then add the numerators. Nine add 20 is 29. And so, the probability of 𝐴 occurring is twenty-nine forty-fifths.
Now, in fact, we could go back to our earlier information and do a little check. We said that the probability of 𝐵 must be less than or equal to the probability of 𝐴. Well, four-ninths is indeed less than twenty-nine forty-fifths. So, this part is certainly true.