Question

In a television network, a survey of 2,500 people was made to know the audience of a debate
and from a movie that aired at different times: 2,100 watched the movie, 1,500 watched the debate and 350
They didn't watch either program. If we randomly choose one of the respondents:

a) What is the probability that I saw the movie and the debate?

b) What is the probability that he saw the movie, knowing that he did not see the debate?

c) Knowing that he saw the movie, what is the probability that he saw the debate?

Answer 1

Let us define the following events:

M: a person watched the movie; and

D: a person watched the debate.

Now, we are given the following probabilities:

P(M) = 2100/2500 = 21/25 ......................(1) [Since, 2100 watched the movie out of the total 2500]

P(D) = 1500/2500 = 3/5 .....................(2) [Since, 1500 watched the debate out of the total 2500]

P(M^cD^c) = 350/1500 = 7/30 [Since, 350 didn't watch neither the movie nor the debate out of the total 2500]

a)

The probability that a randomly chosen respondent saw both the movie and the debate is given by P(MD).

Now, consider the following identity:

Thus, the probability that a randomly chosen respondent saw both the movie and the debate is P(MD) = 101/150 = 0.673333. [ANSWER]

b)

The probability that a randomly chosen respondent saw the movie, knowing that he did not see the debate is given by:

c)

Knowing that a randomly chosen respondent saw the movie, the probability that he saw the debate is given by:

For any queries, feel free to comment and ask.

If the solution was helpful to you, don't forget to upvote it by clicking on the 'thumbs up' button.