Question

About \(26 \%\) of movies coming out of Hollywood are comedies, Warner Bros has been the lead studio for about \(13 \%\) of recent movies, and about \(3 \%\) of recent movies are comedies from Warner Bros. \(^{2}\) Let \(\mathrm{C}\) denote the event a movie is a comedy and \(W\) denote the event a movie is produced by Warner Bros. (a) Write probability expressions for each of the three facts given in the first sentence of the exercise. (b) What is the probability that a movie is either a comedy or produced by Warner Bros? (c) What is the probability that a Warner Bros movie is a comedy? (d) What is the probability that a comedy has Warner Bros as its producer? (e) What is the probability that a movie coming out of Hollywood is not a comedy? (f) In terms of movies, what would it mean to say that \(\mathrm{C}\) and \(\mathrm{W}\) are disjoint events? Are they disjoint events? (g) In terms of movies, what would it mean to say that \(\mathrm{C}\) and \(\mathrm{W}\) are independent events? Are they independent events?

Short answer

The results are: (a) P(C) = 26/100, P(W) = 13/100, P(C ∩ W) = 3/100. (b) P(C ∪ W) = 0.36. (c) P(C | W) ≈ 0.231. (d) P(W | C) ≈ 0.115. (e) P(not C) = 0.74. (f) C and W are not disjoint events. (g) C and W are not independent events.

Step-by-step solution

Understanding the problem

Let C denote the event a movie is a comedy, W denote the event a movie is produced by Warner Bros. The following are given: P(C) = 0.26, P(W) = 0.13, and P(C ∩ W) = 0.03. (a), (b), (c), (d), and (e) seek to identify probabilities of several events while (f) and (g) analyze the relationship between the two events mentioned, C (comedies) and W (produced by Warner Bros).

Calculation of probability expressions

(a) P(C) = 26/100, P(W) = 13/100, and P(C ∩ W) = 3/100.

Probability of movie being a comedy or produced by Warner Bros

(b) P(C ∪ W) = P(C) + P(W) - P(C ∩ W) = 0.26 + 0.13 - 0.03 = 0.36.

Probability that a Warner Bros movie is a comedy

(c) P(C | W) = P(C ∩ W) / P(W) = 0.03/0.13 ≈ 0.231.

Probability that a comedy has Warner Bros as its producer

(d) P(W | C) = P(C ∩ W) / P(C) = 0.03/0.26 ≈ 0.115.

Probability that a movie is not a comedy

(e) P(not C) = 1 - P(C) = 1 - 0.26 = 0.74.

Interpreting disjoint events

(f) If C and W are disjoint events, it would mean that no comedy is produced by Warner Bros. They are not disjoint, since P(C ∩ W) = 0.03 > 0.

Interpreting independent events

(g) If C and W are independent events, it would mean that the outcome of one does not affect the outcome of the other. They are not independent, since P(C ∩ W) ≠ P(C)P(W).