On average 68 % of Finite Mathematics students spend some time in the Mathematic

Question

On average 68 % of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students in the class were asked how many minutes per week they spent in the resource room and whether they passed or failed. The passing rates are summarized in the following table:

If a randomly chosen student did not pass the course, what is the probability that he or she did not study in the resource room?

Explanation / Answer

Let S denote the event that a student passed.

Let H, M, L denote the events that the student spent “more than 90 mins”, “between 1 and 90 mins” and “None” respectively.

We are given P(H) + P(M) = 0.68. P(H) = 0.5 × 0.62 = 0.34.

So, P(M) = 0.34, P(L) = 1 0.68 = 0.32.

We are also given the following values: P(S|L) = 0.25, P(S|M) = 0.52, P(S|H) = 0.81.

We are asked to find P(L|S¯).

P(S) = P(S L) + P(S M) + P(S H)

= P(S|L) × P(L) + P(S|M) × P(M) + P(S|H) × P(H)

= 0.25 × 0.32 + 0.52 × 0.34 + 0.81 × 0.34 = 0.5322.

P(S¯) = 1 P(S) = 0.4678.

And, P(L S¯) = P(L) P(L S) = 0.32 0.25 × 0.32 = 0.24.

Combining, we have,

P(L|S¯) = P(L S¯)/P(S¯)

=0.24/0.4678

=0.5130

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