The probability that a married man watches a certain television show is 0.4 and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that his wife does, is 0.7. Find the probability that
a) a married couple watches the show
b) a wife watches the show given that her husband does;
c) at least 1 person of a married couple will watch the show.
09036085282
0.55
Let's solve the questions step-by-step:
a) To find the probability that a married couple watches the show, we need to find the probability that both the husband and wife watch the show. Since the information about the probability of a husband watching the show is not given directly, we can use the concept of conditional probability.
Let's assume the probability that a husband watches the show is denoted by P(H), and the probability that a wife watches the show is denoted by P(W).
We are given that P(W) = 0.5, and P(H|W) = 0.7 (probability of a man watching the show given that his wife does).
The probability that both the husband and wife watch the show is given by P(H and W) = P(H|W) * P(W).
= 0.7 * 0.5
= 0.35
Therefore, the probability that a married couple watches the show is 0.35.
b) To find the probability that a wife watches the show given that her husband does, we need to find the conditional probability P(W|H).
Using Bayes' theorem, we can write:
P(W|H) = (P(H|W) * P(W)) / P(H)
We are given that P(H|W) = 0.7, P(W) = 0.5, and P(H) is not provided.
Without the value of P(H), we cannot calculate P(W|H) at this point.
c) To find the probability that at least one person in a married couple will watch the show, we can use the concept of complementary probability.
P(at least one person watches the show) = 1 - P(neither person watches the show)
We are given that P(H) = 0.4 (probability of a man watching the show), and P(W) = 0.5 (probability of a wife watching the show).
The probability that neither person watches the show is given by P(neither person watches the show) = (1 - P(H)) * (1 - P(W))
= (1 - 0.4) * (1 - 0.5)
= 0.6 * 0.5
= 0.3
Therefore, the probability that at least one person in a married couple will watch the show is 1 - 0.3 = 0.7.
To find the probability in each scenario, we will use conditional probability and basic probability rules. Let's tackle each question one by one:
a) Probability that a married couple watches the show:
To find this probability, we need to calculate the joint probability that both the husband (H) and the wife (W) watch the show. We denote this as P(H ∩ W).
Using conditional probability, P(H ∩ W) = P(H | W) * P(W) (Equation 1)
From the given information, we know that P(H | W) = 0.7 and P(W) = 0.5.
Substituting these values into Equation 1, we get:
P(H ∩ W) = 0.7 * 0.5 = 0.35
Therefore, the probability that a married couple watches the show is 0.35.
b) Probability that a wife watches the show given that her husband does:
To find this probability, we need to calculate the conditional probability of the wife watching the show (W), given that the husband (H) watches the show. We denote this as P(W | H).
Using conditional probability, P(W | H) = P(H ∩ W) / P(H) (Equation 2)
From our previous calculation, we found P(H ∩ W) = 0.35. We are not given the probability of a man watching the show (P(H)), but we can calculate it using the provided information.
P(H) = P(H | W) * P(W) + P(H | ~W) * P(~W) (Equation 3)
P(H) = 0.7 * 0.5 + P(H | ~W) * (1 - 0.5)
We are missing one piece of information to calculate P(H | ~W), which is the probability that a man watches the show, given that his wife does not watch it. Without this information, we cannot proceed and find the probability of a wife watching the show given that her husband does.
c) Probability that at least one person of a married couple will watch the show:
To find this probability, we need to calculate the probability of either the husband (H) or the wife (W), or both, watching the show.
Using basic probability rules, P(H ∪ W) = P(H) + P(W) - P(H ∩ W) (Equation 4)
From our previous calculations, we know that P(H) = 0.35, P(W) = 0.5, and P(H ∩ W) = 0.35.
Substituting these values into Equation 4, we get:
P(H ∪ W) = 0.35 + 0.5 - 0.35 = 0.5
Therefore, the probability that at least one person of a married couple will watch the show is 0.5.