If the probability of 𝐴 given 𝐵 equals one-third and the probability of 𝐵 equals twelve twenty-fifths, find the probability of 𝐴 intersect 𝐵.
Firstly, let’s just clarify some of the notation used in this question. And this first piece here, which is a probability of 𝐴 and then a vertical line and then 𝐵, I read this as the probability of 𝐴 given 𝐵. And it just means the conditional probability of the event 𝐴 occurring given that we already know the event 𝐵 has occurred.
Secondly, this piece of notation here, which I read as the probability of 𝐴 intersect 𝐵. On a Venn diagram, the intersection of two sets 𝐴 and 𝐵 is their overlap. So, when we talk about the probability of 𝐴 intersect 𝐵, we mean the probability that events 𝐴 and 𝐵 both happen.
We have a formula that connects the three probabilities that we’re given in the question. The probability of 𝐴 and 𝐵 both occurring can be found by finding the probability that 𝐵 occurs and then multiply it by the probability of 𝐴 given 𝐵. That’s the probability that 𝐴 occurs given that 𝐵 has already occurred.
Dividing both sides of this formula through by the probability of 𝐵 gives a rearrangement. The probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersect 𝐵 divided by the probability of 𝐵. And this is called the conditional probability formula. And it gives us a definition for the probability of 𝐴 given 𝐵.
However, in this question, we are asked to find the probability of 𝐴 intersect 𝐵. So, we can use the first version of this formula that we wrote down. The probability of 𝐵 is twelve twenty-fifths. And the probability of 𝐴 given 𝐵 is one-third. So, we have that the probability of 𝐴 intersect 𝐵 is twelve twenty-fifths multiplied by one-third. We can simplify these fractions by cross-cancelling a factor of three from the 12 in the numerator of the first fraction and the three in the denominator of the second to give four twenty-fifths multiplied by one over one. This is just equal to four twenty-fifths. So, the probability of 𝐴 intersect 𝐵 for the events with the given probabilities is four twenty-fifths.
Now just a more general comment, before you learned about conditional probability, you probably learned that if events 𝐴 and 𝐵 are independent, then to find the probability of them both happening, so that’s the probability of their intersection, we can multiply the individual probabilities together. So, we just multiply the probability of 𝐵 by the probability of 𝐴.
The conditional probability formula is really just an extension of this, to situations where the events 𝐴 and 𝐵 are not independent. We still have the probability of 𝐵, but we replace the probability of 𝐴 with the probability of 𝐴 given 𝐵. Because the probability for the second event will be affected by the outcome of the first.
At any rate, if the events 𝐴 and 𝐵 are independent, then the probability of 𝐴 given 𝐵 will just be equal to the probability of 𝐴. So, our conditional probability formula is just a more general version of the formula we use when the events are independent. We found that, in this question, the probability of 𝐴 intersect 𝐵 is four twenty-fifths. This fraction can’t be simplified any further, as the numerator and denominator have no common factors other than one.