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. Let C denote the event a movie is a comedy and W denote the event a movie is produced by Warner Bros.

Required:
a. What is the probability that a movie is either a comedy or produced by Warner Bros?
b. What is the probability that a Warner Bros movie is a comedy?
c. What is the probability that a comedy has Warner Bros as its producer?

Answer 1

a. 0.36 = 36% probability that a movie is either a comedy or produced by Warner Bros.

b. 0.2308 = 23.08% probability that a Warner Bros movie is a comedy

c. 0.1154 = 11.54% probability that a comedy has Warner Bros as its producer

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

a. What is the probability that a movie is either a comedy or produced by Warner Bros?

We solve this question treating these events as Venn sets.

I am going to say that:

Set A: Movie is a comedy.

Set B: Movie is produced by Warner Bros.

26% of movies coming out of Hollywood are comedies

This means that
`P(A) = 0.26`

Warner Bros has been the lead studio for about 13% of recent movies

This means that
`P(B) = 0.13`

3% of recent movies are comedies from Warner Bros.

This means that
`P(A \cap B) = 0.03`

The desired probability is:

`P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.26 + 0.13 - 0.03 = 0.36`

0.36 = 36% probability that a movie is either a comedy or produced by Warner Bros.

b. What is the probability that a Warner Bros movie is a comedy?

Conditional probability, in which

Event A: Produced by Warner Bros

Event B: Comedy

Warner Bros has been the lead studio for about 13% of recent movies

This means that
`P(A) = 0.13`

3% of recent movies are comedies from Warner Bros.

This means that
`P(A \cap B) = 0.03`

The desired probability is:

`P(B|A) = (P(A \cap B))/(P(A)) = (0.03)/(0.13) = 0.2308`

0.2308 = 23.08% probability that a Warner Bros movie is a comedy.

c. What is the probability that a comedy has Warner Bros as its producer?

Now,

Event A: Comedy

Event B: Warner Bros as its producer.

About 26% of movies coming out of Hollywood are comedies

This means that
`P(A) = 0.26`

3% of recent movies are comedies from Warner Bros.

This means that
`P(A \cap B) = 0.03`

The desired probability is:

`P(B|A) = (P(A \cap B))/(P(A)) = (0.03)/(0.26) = 0.1154`

0.1154 = 11.54% probability that a comedy has Warner Bros as its producer