490 CHAPTER 8 Discrete Probability
- For the two nickels and the dime in Exercise 8, there are eight possible combinations
of heads and tails: tails on all coins; heads on nickel 1, tails on nickel 2, tails on the
dime; and so on.
(a) Assuming that each of these eight combinations is equally likely, what probabil-
ity density should be assigned to the sample space Q2 of Exercise 8? Specify the
probability density by giving its value on each 0 E QŽ.
(b) Invent a new sample space Q such that each of the situations in Exercise 8 can
be described as an event, and specify a reasonable probability density for the new
sample space.
(c) Describe each of the situations in Exercise 8 as an event in the new sample space
from part (b).
(d) Calculate the probability of each of the events in part (c) using the new probability
density.
10. Consider a sample space 2 = {a, e, i, o, u} endowed with the following probabil-
ity density: p(a) = 0.22, p(e) = 0.35, p(i) = 0.13, p(o) = 0.20, and p(u) = 0.10.
Determine the probabilities of the following events:
(a) {a, ol
(b) 0
(c) The event E consisting of all those outcomes in 02 that come after the letter k in
the alphabet- A small zoo records the proportions of visitors who prefer various animals as their
favorites. Suppose that the elephants are preferred by 15%, the monkeys by 25%, the
polar bears by 30%, the seals by 20%, and the boa constrictors by 10%. Suppose we
are going to select a visitor at random and ask what animal that person prefers.
(a) Set up a sample space QŽ, and define a probability density on it using the given
data.
(b) Reformulate the descriptions of the following events as subsets of S2:
i. The preferred animal has four legs
ii. The preferred animal has legs
iii. The preferred animal has either a trunk or flippers.
(c) Calculate the probability of each event in part (b). - A fair coin is tossed five times. Determine the probability that:
(a) It turns up tails every time.
(b) It turns up heads at most three times.
(c) It turns up heads twice in a row exactly one time. - Suppose that five names are drawn from a hat at random and listed in the order in
which they are drawn. A name on the list is "out of order" if its position is not the
same as its position after the list has been alphabetized. Determine the probability of
each of the following:
(a) Exactly two names are out of order.
(b) At least two names are out of order.
(c) Exactly one name is out of order. - What is the probability that a card drawn at random from a 52-card deck will be a heart
or an even-numbered card?