518 CHAPTER 8 Discrete Probability
- Suppose that E 1 , E 2 ... , Ek are events in the same sample space and that some pair
Ei, Ej of these events are disjoint.
(a) If all the events have positive probability, can the set {El, E2, ... , Ekd be an in-
dependent set of events? Explain your answer.
(b) If one or more of the events has 0 probability, can the set {E 1 , E2. Ek be an
independent set of events? - A fair penny and a fair nickel are tossed. Let A be the event "heads on the penny." Let
B be the event "tails on the nickel." Let C be the event "the coins land the same way."
(a) Choose a sample space Q2, and represent A, B, and C in terms of Ž2.
(b) Which pairs of events chosen from A, B, and C are independent pairs?
(c) Is the set of events {A, B, C} an independent set? - Suppose that QŽ is a sample space with a probability density function p, and suppose
that A C Q2. Let P(A) denote the probability of A. Assume that P(A) > 0. Define a
function pI on A as follows: Forf) E A, PI0(w) = p(w))/P(A).
(a) Show that if w, 0)2 E A and p(wl), P((02) # 0, then
P(0I) Pl(0w))
P(a)2) Pl (a)2)(b) Show that if B and C are nonempty subsets of A with elements that have positive
probabilities, thenP(B) _ P 1 (B)
P(C) Pi(C)(c) Show that PI is a probability density function on 2 1 = A.- Suppose we have two coins. One is fair, but the other one has two heads. We choose
one of them at random and flip it. It comes up heads.
(a) What is the probability the coin is fair?
(b) Suppose we flip the same coin a second time. What is the probability that it comes
up heads?
(c) Suppose the coin comes up heads when flipped the second time. What is the prob-
ability the coin is fair? - A television show features the following weekly game: A sports car is hidden behind
one door, and a goat is hidden behind each of two other doors. The moderator of the
show invites the contestant to pick a door at random. Then, by tradition, the moderator
is obligated to open one of the two doors not chosen to reveal a goat (there are two
goats, so there is always such a door to open). At this point, the contestant is given the
opportunity to stand pat (do nothing) or to choose the remaining door. Suppose you are
the contestant, and suppose you prefer the sports car over a goat as your prize. What
do you do? (Hint: It may help to model this as a two-stage dependent trials process,
but it may not be obvious how to do this).