About 20 of movies recently coming out of Hollywood are come

About 20% of movies recently coming out of Hollywood are comedies. Warner Bros. has been the lead studio for about 9% of recent movies. About 2% of recent movies are Warner Bros. comedies. Let C denote the event a recent movie is a comedy and W denote the event a recent movie is produced by Warner Bros. Assume that all movies mentioned below are recent movies coming out of Hollywood.

Write probability expressions for the three facts given above (in the first three sentences).

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

If a recent movie was produced by Warner Bros., what is the probability that it is a comedy?

What proportion of recent comedies are produced by Warner Brothers?

What is the probability that a recent Hollywood movie is not a comedy?

Are C and W mutually exclusive (disjoint) events? Justify your answer using probability.

Are C and W independent or dependent events? Justify your answer using probability.

Solution

Three facts

About 20% of movies recently coming out of Hollywood are comedies , P(C) = 0.2

Warner Bros. has been the lead studio for about 9% of recent movies, P(W) = 0.09

About 2% of recent movies are Warner Bros. comedies, P(C n W) = 0.02

a.) probability that a recent movie is a comedy or is produced by Warner Bros. = P(C u W)

= P(C) + P(W) - P(C n W) = 0.2 + 0.09 - 0.02 = 0.27

b.) If a recent movie was produced by Warner Bros., the probability that it is a comedy = P(C|W)

= P(C n W)/P(W) = 0.02/0.09 = 2/9 = 0.2222

c.) proportion of recent comedies are produced by Warner Brothers = P(W|C)

= P(C n W)/P(C) = 0.02/0.2 = 0.1

d.) the probability that a recent Hollywood movie is not a comedy = P(C^c) = 1 - P(C) = 0.8

e.) P(C n W) = 0.02 is not zero, so C and W are not mutually exclusive events.

f.) P(C n W) = 0.02

P(C)*P(W) = 0.2*0.09 = 0.018

SO P(C n W) is not equal to P(C)*P(W)

So C and W are not independent events

SO C and W are dependent events