Exercises 489
w0 p1 (w0) P2 (W0)
Woi -1/2 1/31f7
(02 1/4 2/31/7-
W03 1/4 1/3/7
a04 1/2 2/3/7
(05 1/2 (,/7--2)/17
- Two nickels and a dime are shaken together and thrown. All the coins are fair. We
are allowed to keep the coins that turn up heads. Give two sample spaces together
with probability density functions that reasonably describe this situation. Explain your
answer.
- A penny, a nickel, and a dime are shaken together and thrown. Someone proposes
the sample space 0Ž = {1, 5, 10), explaining that the outcomes represent the values of
the coins. Then, the person suggests the probability density function p(l) = p(5) =
p(10) = 1/2, explaining that since the coins are fair, a uniform probability density
is appropriate. Comment on several aspects of this situation. Do you agree with this
model?
- A penny, a nickel, and a dime are shaken together and thrown. Suppose that the nickel
turns up heads twice as frequently as the penny and that the dime turns up heads half
as frequently as the nickel. Define a sample space, and using the frequency interpreta-
tion, assign a reasonable probability density function based on the assumption that the
penny is a fair coin.
- Given the sample space
Q = {0, 5, 10, 15, 201
which of the following events are in the sample space?
(a) (5, 101
(b) 10, 5, 10, 15, 20)
(c) 0
(d) 0
(e) {0}
(f) {5)
- Two nickels and a dime are shaken together and thrown. We are allowed to keep the
coins that turn up heads. We choose a sample space 02 = {0, 5, 10, 15, 20), the out-
comes of which correspond to the amounts that we can keep. For each of the following
situations, either describe the situation as an event in^02 by listing the elements in the
appropriate subset of Q2 or state that the situation cannot be described as an event in
this particular sample space:
(a) No heads.
(b) All heads.
(c) Exactly one coin turns up heads.
(d) Exactly one of the nickels turns up heads.
(e) The dime turns up heads.
