In the final exams, 40 percent of the students failed chemistry, 25 percent failed physics, and 19 percent failed both chemistry and physics. What is the probability that a randomly selected student failed physics given that he passed chemistry?
The words “given that” in the question indicate that we are dealing with conditional probability. One way of answering a question of this type is using a Venn diagram. The two circles inside our Venn diagram represent the number of students that failed chemistry and failed physics. We are told that 19 percent of students failed both chemistry and physics. As a decimal, 19 percent is equal to 0.19, so we can place this in the intersection of the two circles on our Venn diagram. We are told that 25 percent of students failed physics. As a decimal, 25 percent is equal to 0.25. As we have already included 19 percent out of the 25 percent, we need to subtract 0.19 from 0.25. This is equal to 0.06, telling us that six percent of the students failed only physics.
We can repeat this process for chemistry, and we know that 40 percent of the students failed this subject. 0.4 or 0.40 minus 0.19 is equal to 0.21. This means that 21 percent of the students failed only chemistry. The three values in our Venn diagram currently sum to 0.46. As we know that probabilities sum to one, we need to subtract 0.46 from one. This is equal to 0.54. 54 percent of the students did not fail either exam. This is the same as passing both physics and chemistry.
In this question, we need to find the probability that the student failed physics given that he passed chemistry. The students that passed chemistry will be those outside of the chemistry circle. We have two values here, 0.06 and 0.54. These sum to give us 0.60 or 0.6. 60 percent of the students passed chemistry, which ties in with the fact that 40 percent failed.
We can see from the Venn diagram that six percent of this 60 percent failed physics. The probability that a student failed physics given that they passed chemistry is therefore equal to 0.06 out of 0.6. We can multiply both the numerator and denominator by 100, giving us six over 60. This in turn simplifies to one over 10 or one-tenth. We can therefore conclude that the probability that a randomly selected student failed physics given that he passed chemistry written as a decimal is 0.1.
This leads us to a general formula we can use when dealing with conditional probability. The probability that event 𝐴 happens given that event 𝐵 happens is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. In this question, the numerator was the probability that a student failed physics and passed chemistry. This was equal to 0.06. The denominator was the probability that the student passed chemistry. This was equal to 0.6. Dividing these gave us our answer of 0.1.