Final answer:
To calculate the fraction of students who failed and went partying, we use the fact that 20% of students partied and are twice as likely to fail. We don't need the exact failure rate to find that 40% of the students who failed had partied before the exam.
Step-by-step explanation:
To find the fraction of students who failed and went partying, we need to use the information given about their likelihood of failing. Let's define the following variables:
We know that 20% of the students partied before the exam, so P(Party) = 0.20. The students who party are twice as likely to fail, so we could write P(Fail|Party) = 2 × P(Fail). However, we do not have the exact probability of failing P(Fail), and indeed we don't need it for this question. What we need to calculate is the portion of failed students that partied. This is given by:
P(Party|Fail) = (P(Fail|Party) × P(Party)) / P(Fail)
By substituting the proportion of students who partied (0.20) and observing that students who party are twice as likely to fail, we can infer that:
P(Fail|Party) is twice as much as the overall failing rate, which we can call 2P(Fail).
Since the probability of partying given a failure is the probability of partying and failing divided by the probability of failing in general, we have:
P(Party|Fail) = (2P(Fail) × 0.20) / P(Fail) = 2 × 0.20 = 0.40
So, 40% of the students who failed went partying.