Question Video: Using Venn Diagrams to Calculate Dependent Probabilities Mathematics • 10th Grade

The figure shows a Venn diagram with some of the probabilities given for two events 𝐴 and 𝐵. Work out the probability of 𝐴 intersection the complement of 𝐵. Work out 𝑃(𝐴). Work out 𝑃(𝐵 ∣ 𝐴).

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Video Transcript

The figure shows a Venn diagram with some of the probabilities given for two events 𝐴 and 𝐵. Work out the probability of 𝐴 intersection the complement of 𝐵. Work out the probability of 𝐴. Work out the probability of 𝐵 given 𝐴.

Let’s begin by having a quick look at the Venn diagram and seeing what some of these values represent. We have two events labeled 𝐴 and 𝐵. And in the center, we’ve got this value of 0.2, which represents the probability of the event 𝐴 intersection 𝐵. In other words, it’s the probability of event 𝐴 and 𝐵 occurring. The value of 0.4 represents an event being 𝐵 but not 𝐴. Notice that outside of the circles for 𝐴 and 𝐵, we have this value of 0.3. This represents the probability of neither 𝐴 nor 𝐵 occurring. So let’s take a look at the first question to work out the probability of 𝐴 intersection the complement of 𝐵. This bar represents the complement, and it really means not 𝐵.

When we’re finding the probability of 𝐴 intersection the complement of 𝐵, it’s equivalent to finding the probability that 𝐴 occurs but 𝐵 does not. It would look like this on the Venn diagram. But how exactly do we calculate a probability? Well, we’ll need to remember a very important fact about probabilities. The sum of the probabilities of all outcomes must equal one. In other words, something has to happen. Whether the outcome is event 𝐴 or 𝐵 or both or neither, all of these probabilities will add to give us one. In fact, the only probability that’s missing on our diagram is that of the probability of 𝐴 intersection not 𝐵. Therefore, if we add together 0.2, 0.4, and 0.3 and subtract these from one, then we’ll find our missing probability. So the probability of 𝐴 intersection the complement of 𝐵 is 0.1.

Let’s have a look at the second question to work out the probability of 𝐴. When it comes to answering a question like this, it can be very tempting just to give the value of 0.1. However, remember that 0.1 represents event 𝐴 and not 𝐵, and so we also need to include those events which are 𝐴 and 𝐵, in other words, everything that’s in the circle for event 𝐴. Adding the probabilities in of 0.1 and 0.2 gives us 0.3. And so the answer for the second part of this question is that the probability of 𝐴 is equal to 0.3.

In the final part of this question, we’re asked to work out a conditional probability, the probability of 𝐵 given 𝐴. We should remember the formula for conditional probability that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. Before we rush to apply this formula, however, there’s a change that we’ll need to make. This formula allows us to calculate the probability of 𝐴 given 𝐵, but we actually need to work out the probability of 𝐵 given 𝐴. We’ll, therefore, need to switch our values of 𝐴 and 𝐵 in the formula.

Notice that when we’ve done so, the numerator of our fraction is the probability of 𝐵 intersection 𝐴. The probability of 𝐵 intersection 𝐴 is actually the same as the probability of 𝐴 intersection 𝐵. It still refers to this portion which is both event 𝐴 and event 𝐵. Let’s now plug in the values to find our conditional probability. The section that’s highlighted is the probability of 𝐵 intersection 𝐴, and remember that we calculated in our second question that the probability of 𝐴 is 0.3. We can simplify this fraction by multiplying the numerator and denominator by 10. So the answer for the third part of the question is that the probability of 𝐵 given 𝐴 is two-thirds.

We can think about this visually on the Venn diagram by remembering that it’s given that we have event 𝐴, then what’s the probability that it’s also event 𝐵. So it have the proportion of 0.2 out of 0.3, which is the fraction two-thirds.