Question Video: Using Venn Diagrams to Calculate Dependent Probabilities Mathematics

Video Transcript

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

We recall that in any Venn diagram, the probabilities must sum to one. 0.3 plus 0.2 plus 0.4 plus 0.1 is equal to one. The first part of our question asks us to calculate the probability of 𝐴. This is equal to the sum of all the probabilities inside circle 𝐴. We need to add 0.3 and 0.2. This is equal to 0.5. The probability of event 𝐴 occurring is 0.5.

The second part of our question wants us to calculate the probability of 𝐴 and 𝐵. This is known as the intersection. It is the part of the Venn diagram where both 𝐴 and 𝐵 occur. The probability in the overlap of the two circles is 0.2. This means that the probability of 𝐴 intersection 𝐵 is 0.2.

The final part of our question wants us to calculate the probability of 𝐵 given that 𝐴 occurs. The notation in this question means given that. We recall that the probability of 𝐵 given 𝐴 is equal to the probability of 𝐵 intersection 𝐴 divided by the probability of 𝐴. We have already worked out both of these answers. It’s important to note that the probability of 𝐵 intersection 𝐴 is the same as the probability of 𝐴 intersection 𝐵. The probability of 𝐵 given 𝐴 is, therefore, equal to 0.2 divided by 0.5. This is equal to two-fifths or 0.4. The probability of 𝐵 given 𝐴 is 0.4.

We could also have worked this out from our Venn diagram. As we want the probability of 𝐵 given that 𝐴 happens, we begin with the 0.5 inside circle 𝐴. Which of these probabilities is also in circle 𝐵? This is the 0.2. So once again we have 0.2 out of or divided by 0.5. Our three answers in this question are 0.5, 0.2, and 0.4. The 0.1 that was on the outside of both circles is the probability that neither event 𝐴 nor event 𝐵 occur.