Question

55. A family has three children. Let A be the event that they have less than two girls and B be the event that they have exactly two girls.

(a) List all of the basic outcomes in A.
(b) List all of the basic outcomes in B.
(c) List all of the basic outcomes in A ∩ B
(d) List all of the basic outcomes in A U B.
(e) If male and female births are equally likely, what is the probability of A?

56. Let B = A^c. Are A and B mutually exclusive? Are they collectively exhaustive?

67. Suppose that P(B) = 0.4, P(A|B) = 0.1 and P(A|B^c) = 0.9

(a) Calculate P(A)

(b) Calculate P(A|B)

71. Suppose a couple decides to have three children. Assume that the sex of each child is independent, and the probability of a girl is 0.48, the approximate figure in the US.

(a) How many basic outcomes are there for this experiment? Are they equally likely?

(b) What is the probability that the couple has at least one girl?

72. Let A and B be two arbitrary events. Use the addition rule and axioms of probability to establish the following results.

(a) Show that P(A U B) ≤ P(A) + P(B).

(this is called Boole’s inequality).

(b) Show that P(A n B) is greater than or equal to P(A) + P(B) - 1.

(this is called Bonferroni's Inequality)

73. Let A and B be two mutually exclusive events such that P(A) > 0 and P(B) > 0. Are A and B independent?

104. A multiple-choice quiz has 12 questions, each of which has 5 choices. To pass you need to get at least 8 of them correct. Nina forgot to study, so she simply guesses at random.

1. Let the random variable X denote the number of questions that Nina gets correct on the quiz. What kind of random variable is X? Specify all parameter values.

2. Calculate the probability that Nina passes the quiz.

Answer 1

Let us answer question 55.

Here, the family has 3 children. So, the sample space is given by, S={GGG, GBB,BGB,BBG,GGB,GBG,BGG,BBB}, where G denote girl child and B denote boy child. There can be three girls or two girls or one girl or no girl.

Let A be the event that less than two girls. That is, there are 0 girls or 1 girls.

(a) From the sample space, the outcomes favorable to A is given by, A={BBB,GBB,BGB,BBG}

Let B denote the event that exactly two girls. That is, the number of girls is 2.

(b) From the sample space, the outcome favorable to B is given by, B={GGB,GBG,BGG}

(c) We have to list the outcomes favorable to A ∩ B. That is, the outcomes common to both A and B.

From the answers(a) and (b),we can see that there is no common outcome in A and B. So the answer is

A ∩ B={ }

(d)We have to list the outcomes favorable toA U B. That is, we have list all the outcomes either in A or in B or in both. So, the outcomes favorable to A U B is given by, A U B={BBB,GBB,BGB,BBG,GGB,GBG,BGG}