506 CHAPTER 8 Discrete Probability
- A coin that is twice as likely to show heads than it is tails is tossed three times.
(a) Describe this experiment as a Bernoulli process.
(b) Use a tree diagram to assign a probability density. - Suppose the coin in Exercise 9 is tossed n times where n is some arbitrary positive
integer.
(a) Sketch a tree diagram for this experiment.
(b) Give a formula for the number of outcomes in which heads occur exactly k times.
(c) Give a formula for the value of the probability density function on each of the
outcomes described in part (b).
(d) What would be the formula for the value of the probability density function on
each of the outcomes in part (b) if the coin were fair? - A coin that is twice as likely to show heads than it is tails is tossed three times. Suppose
we are only interested in the number of heads.
(a) Formulate this experiment in terms of a sample space with outcomes that give the
number of heads.
(b) Assign a probability density function to this sample space.
(c) Describe the event "at least two heads" as a set of outcomes in this sample space.
(d) What is the probability of the event in part (c)? - Consider the events described in Exercise 6 regarding Example 2. Which (if any) of
these events can be expressed in cross product form? Give the cross product expression
if one exists. - For the cross product sample space found in Exercise 8(a), give the cross product
formulation, and calculate the probability of each of the following events:
(a) An even number of spots shows on each of the die.
(b) The loaded die shows six spots.
(c) The fair die shows either five or six spots, and the loaded die shows six spots. - For the coin, die, and card experiment of Exercise 7, let the coin have sample space
Q = IH, T), the die have sample space Q22 = {1, 2, 3, 4, 5, 6}, and the card deck
have sample space Q23 = {1, 2,..., 51, 52}.
(a) What event E* in , X Q22 X Qi 3 corresponds to the event E 1 = "heads on the
coin"?
(b) What event E* in Q2, X Q2 X Q23 corresponds to the event E 2 = "4 or higher on
the die?"
(c) Formulate "getting a head on the coin and a 4 or higher on the die" as an event
in Q1 X Q2 X 23 by indicating the outcomes in the event. Then, formulate this in
terms of E* and E*.
(d) Calculate the probability of the event "heads on the coin and a 4 or higher on the
die" in two ways by using each of the formulations of part (c).
(e) Formulate the event "either a head on the coin or a 4 or better on the die" in terms
of E* and E*. What is the probability of this event?- Give two ways you could label the outcomes of an experiment that consists of flipping
a dime three times. Determine a probability density function for each of the two ways. - What is the probability of flipping k heads out of m tosses (0 < k < m)? What are the
numerical results for five heads out of eight tosses?