# Video: Finding the Conditional Probability of an Event

A class contains 100 students. 70 of them like mathematics, 60 like physics, and 40 like both. If a student is chosen at random, using a Venn diagram, find the probability that they like mathematics but not physics.

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### Video Transcript

A class contains 100 students. 70 of them like mathematics, 60 like physics, and 40 like both. If a student is chosen at random, using a Venn diagram, find the probability that they like mathematics but not physics.

We are asked to solve this problem using a Venn diagram. We are told that there are 100 students in total, and we are dealing with two subjects: mathematics and physics. We are told that 40 students like both mathematics and physics. This means that the probability of selecting a student that likes both subjects is 40 out of 100. This is equal to 0.4. In our Venn diagram, we can place the number 40 in the intersection of both circles, as this represents the students that like both subjects.

We are told that 70 of the students like mathematics. This means that the probability of selecting a student that likes mathematics is 70 out of 100 or 0.7. We have already included 40 of these students on our Venn diagram, and 70 minus 40 is equal to 30. This means that in the section that represents just mathematics, we have 30 students. We can repeat this process for physics. There are 60 students that like this subject; therefore, the probability of selecting a student that likes physics is 60 out of 100 or 0.6. As 60 minus 40 is equal to 20, there will be 20 students in the section of the Venn diagram representing only physics.

The three values in our Venn diagram, 30, 40, and 20, sum to 90. As the class contains 100 students in total, there must be 10 students that did not like physics or mathematics. We have been asked to find the probability that a student likes mathematics but not physics. The probability of not liking physics is denoted Ph bar or Ph prime. From our Venn diagram, we see that there are 30 students that like mathematics but not physics. This means that the probability is equal to 30 over 100. Written as a decimal, this is 0.3. The probability that a student randomly chosen from the class likes mathematics but not physics is 0.3.

This leads us to a general formula we can use when dealing with probability. The probability of event 𝐴 occurring and event 𝐵 not occurring written P of 𝐴 intersection 𝐵 bar is equal to the probability of event 𝐴 minus the probability of event 𝐴 intersection 𝐵. In this question, the probability that the student likes maths but not physics is equal to 0.7 minus 0.4. Once again, this gives us the correct answer of 0.3.