## In a concert band, the probability that a member is in the woodwind section is 0.40. The probability that a member plays clarinet, given tha

Question

In a concert band, the probability that a member is in the woodwind section is 0.40. The probability that a member plays clarinet, given that he or she is in the woodwind section, is 0.50. What is the probability that a randomly selected band member is in the woodwind section and plays clarinet?

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4 weeks 2021-09-23T11:15:05+00:00 2 Answers 0

.2

Step-by-step explanation:

Let A represent the probability a band member plays clarinet.

Let B represent the probability a band member is in the woodwind section.

These two events, A and B, are not mutually exclusive, because you can be a clarinet and be in the woodwind section. (Of course! Clarinet gang for life.) They are not independent, because if a band member is a clarinetist, they must be in the woodwind section.

So P(B) = .4

P(A|B) = .5

The question is asking us for the probability a band member is in the woodwind section and plays clarinet, or P(A and B).

We can figure out P(A and B), also known as the intersection, by using this formula:

P(A|B) = P(A and B) / P(B)

.5 = P(A and B) / .4

P(A and B) = .2

In case this was too much algebraic thinking, you can also think about it in your head without using formulas. 40% of the band are woodwinds, and 50% of the woodwinds are clarinets, so 20% of the band are clarinetists who are in the woodwind section. Easy!