Question

In the final exams, 40% of the students failed chemistry, 25% failed physics, and 19% failed both chemistry and physics. What is the probability that a randomly selected student failed physics given that he passed chemistry?

I have answered the question in the image below, but I would like to know if it is correct. If it is not, please include an explanation of why, as well as the step by step to get the correct answer

Answer 1

To solve this problem, we can use conditional probability.

Let's assume that there were 100 students in the final exam.

According to the problem, 40% of the students failed chemistry, which means that 60% of the students passed chemistry.

• We can see that 25% of the students failed physics, and 19% of the students failed both chemistry and physics.

To find the probability that a randomly selected student failed physics given that he passed chemistry, we need to use Bayes' theorem:

`\sf P(Failed\: Physics | Passed\: Chemistry) = (P(Failed\: Physics\: and\: Passed\: Chemistry))/( P(Passed\: Chemistry))`

We already know that P(Failed Physics and Passed Chemistry) = 6 students (from the Venn diagram), and P(Passed Chemistry) = 60 students (since 60% of the students passed chemistry).

Therefore,

`\sf P(Failed\: Physics | Passed\: Chemistry) = (6)/(60) = 0.1\: or\: 10\%`

So the probability that a randomly selected student failed physics given that he passed chemistry is 10%.

Answer 2

10%

Explanation:

If 40% failed chemistry then 60% passed chemistry.

If 19% failed both chemistry and physics, and 25% failed physics, then 6% passed chemistry and failed physics.

If we let p' represent failing physics and c represent passing chemistry, then ...

P(p'|c) = P(p'c)/P(c)

P(p'|c) = 6%/60% = 0.10 = 10%

If the randomly chosen student passed chemistry, the probability is 10% that he failed physics.

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Your answer is correct.