# 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 ๐ด and ๐ต. Work out the probability of ๐ด intersection the complement of ๐ต. Work out ๐(๐ด). Work out ๐(๐ต โฃ ๐ด).

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### Video Transcript

The figure shows a Venn diagram with some of the probabilities given for two events ๐ด and ๐ต. Work out the probability of ๐ด intersection the complement of ๐ต. Work out the probability of ๐ด. Work out the probability of ๐ต given ๐ด.

Letโs begin by having a quick look at the Venn diagram and seeing what some of these values represent. We have two events labeled ๐ด and ๐ต. And in the center, weโve got this value of 0.2, which represents the probability of the event ๐ด intersection ๐ต. In other words, itโs the probability of event ๐ด and ๐ต occurring. The value of 0.4 represents an event being ๐ต but not ๐ด. Notice that outside of the circles for ๐ด and ๐ต, we have this value of 0.3. This represents the probability of neither ๐ด nor ๐ต occurring. So letโs take a look at the first question to work out the probability of ๐ด intersection the complement of ๐ต. This bar represents the complement, and it really means not ๐ต.

When weโre finding the probability of ๐ด intersection the complement of ๐ต, itโs equivalent to finding the probability that ๐ด occurs but ๐ต does not. It would look like this on the Venn diagram. But how exactly do we calculate a probability? Well, weโ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 ๐ด or ๐ต or both or neither, all of these probabilities will add to give us one. In fact, the only probability thatโs missing on our diagram is that of the probability of ๐ด intersection not ๐ต. Therefore, if we add together 0.2, 0.4, and 0.3 and subtract these from one, then weโll find our missing probability. So the probability of ๐ด intersection the complement of ๐ต is 0.1.

Letโs have a look at the second question to work out the probability of ๐ด. 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 ๐ด and not ๐ต, and so we also need to include those events which are ๐ด and ๐ต, in other words, everything thatโs in the circle for event ๐ด. 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 ๐ด is equal to 0.3.

In the final part of this question, weโre asked to work out a conditional probability, the probability of ๐ต given ๐ด. We should remember the formula for conditional probability that the probability of ๐ด given ๐ต is equal to the probability of ๐ด intersection ๐ต divided by the probability of ๐ต. Before we rush to apply this formula, however, thereโs a change that weโll need to make. This formula allows us to calculate the probability of ๐ด given ๐ต, but we actually need to work out the probability of ๐ต given ๐ด. Weโll, therefore, need to switch our values of ๐ด and ๐ต in the formula.

Notice that when weโve done so, the numerator of our fraction is the probability of ๐ต intersection ๐ด. The probability of ๐ต intersection ๐ด is actually the same as the probability of ๐ด intersection ๐ต. It still refers to this portion which is both event ๐ด and event ๐ต. Letโs now plug in the values to find our conditional probability. The section thatโs highlighted is the probability of ๐ต intersection ๐ด, and remember that we calculated in our second question that the probability of ๐ด 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 ๐ต given ๐ด is two-thirds.

We can think about this visually on the Venn diagram by remembering that itโs given that we have event ๐ด, then whatโs the probability that itโs also event ๐ต. So it have the proportion of 0.2 out of 0.3, which is the fraction two-thirds.