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.

(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 C and W are disjoint events? Are they disjoint events?

(g) In terms of movies, what would it mean to say that C and W are independent events? Are they independent events?

Answer 1

Event C = a movie is comedy

Event W = a movie is produced by Warner Bros.

Since 26% of movies are comedy , we can say that if a movie comes out at random the probability that it would be a comedy = P(C) = 0.26

Probability that a movie is comedy and produced by Warner Bros. = 0.03 (3%)

P(a movie is comedy and produced by Warner Bros.) = P(C and W) = 0.03  

Similarly Probability that a movie is produced by Warner Bros. is = P(W) = 0.13

(e) A movie can either be a comedy or not a comedy .

Complementary event of Event C is = Cc   not a comedy movie

Probability of an event and its complement are related :

Probability that a movie is not a comedy = P(Cc ) = 1 - P(C) = 1- 0.26 = 0.74

(f) Disjoint events are those events that can never occur together .

For events C and W to be disjoint means that they do not occur together meaning a movie cannot be a comedy and produced by Warner Bros. at same time or in other words - Warner Bros . can't make a comedy movie .

If both are disjoint than P(C and W) = 0 which is not the case

C and W are not disjoint Events because there are 3% movies that are comedy and produced by Warner Bros.

(g) Independent events are those events that do not affect the probability of occuring of each other in any way.

For C and W to be independent events it would mean that if a movie that comes out is comedy it would not affect the chances of it being a Warner Bros.. production. or if a movie that comes out is Warner Bros. production then it won't change probability of it being a comedy.

We can check if two events are independent or not .

For Two independent events A and B this condition satisfies: P(A and B) = P(A) P(B)

So we check if C and W are independent or not -

P(C) P(W) = 0.26 * 0.13 = 0.0338   P(C and W) =0.03

So C and W are not independent