Assume that [math]E[/math] and [math]F[/math] are two events with positive probabilities. Show that if [math]P(E|F) = P(E)[/math], then [math]P(F|E) = P(F)[/math].

A coin is tossed three times. What is the probability that exactly two heads occur, given that

• the first outcome was a head?
• the first outcome was a tail?
• the first two outcomes were heads?
• the first two outcomes were tails?
• the first outcome was a head and the third outcome was a head?

A die is rolled twice. What is the probability that the sum of the faces is greater than 7, given that

• the first outcome was a 4?
• the first outcome was greater than 3?
• the first outcome was a 1?
• the first outcome was less than 5?

A card is drawn at random from a deck of cards. What is the probability that

• it is a heart, given that it is red?
• it is higher than a 10, given that it is a heart? (Interpret J, Q, K, A as 11, 12, 13, 14.)
• it is a jack, given that it is red?

A coin is tossed three times. Consider the following events

[math]A[/math]: Heads on the first toss.

[math]B[/math]: Tails on the second.

[math]C[/math]: Heads on the third toss.

[math]D[/math]: All three outcomes the same (HHH or TTT).

[math]E[/math]: Exactly one head turns up.

• Which of the following pairs of these events are independent? (1) [math]A[/math], [math]B[/math] (2) [math]A[/math], [math]D[/math] (3) [math]A[/math], [math]E[/math] (4) [math]D[/math], [math]E[/math]
• Which of the following triples of these events are independent? (1) [math]A[/math], [math]B[/math], [math]C[/math] (2) [math]A[/math], [math]B[/math], [math]D[/math] (3) [math]C[/math], [math]D[/math], [math]E[/math]

From a deck of five cards numbered 2, 4, 6, 8, and 10, respectively, a card is drawn at random and replaced. This is done three times. What is the probability that the card numbered 2 was drawn exactly two times, given that the sum of the numbers on the three draws is 12?

A coin is tossed twice. Consider the following events.

[math]A[/math]: Heads on the first toss.

[math]B[/math]: Heads on the second toss.

[math]C[/math]: The two tosses come out the same.

• Show that [math]A[/math], [math]B[/math], [math]C[/math] are pairwise independent but not independent.
• Show that [math]C[/math] is independent of [math]A[/math] and [math]B[/math] but not of [math]A \cap B[/math].

Let [math]\Omega = \{a,b,c,d,e,f\}[/math]. Assume that [math]m(a) = m(b) = 1/8[/math] and

[math]m(c) = m(d) = m(e) = m(f) = 3/16[/math]. Let [math]A[/math], [math]B[/math], and [math]C[/math] be the events [math]A = \{d,e,a\}[/math], [math]B = \{c,e,a\}[/math], [math]C = \{c,d,a\}[/math]. Show that [math]P(A \cap B \cap C) = P(A)P(B)P(C)[/math] but no two of these events are independent.

What is the probability that a family of two children has

• two boys given that it has at least one boy?
• two boys given that the first child is a boy?

In Example, we used the Life Table (see Appendix C) to compute a conditional probability. The number 93,53 in the table, corresponding to 40-year-old males, means that of all the males born in the United States in 1950, 93.753% were alive in 1990. Is it reasonable to use this as an estimate for the probability of a male, born this year, surviving to age 40?