Due date: Sunday 9/14, 11:59pm
Repeated trials
01
05
Winning the lottery
Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.
If you purchase 20 of these game cards, what is the probability you will win at least once?
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02
06
Watching the Superbowl
A representative from Nielsen ratings randomly selects people in Charlottesville, VA and asks them whether they watched the Superbowl. The probability that any individual in Charlottesville watched the Superbowl is 0.3.
(a) What is the probability that if the representative asks 10 people, that less than 2 of them will have watched the Superbowl?
(b) What is the probability that the representative will have to ask at least 3 people to find someone who watched the Superbowl?
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13
(a)
(b)
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Reliability
03
02
Enough staff to open
A small restaurant needs a minimum number of staff to open: 1 manager, 1 cook, 3 servers, and 1 host. Suppose there are 2 managers, 3 cooks, 3 servers, and 1 host. Each staff member is available with probability 0.95, and their availability is independent of others. What is the probability that the restaurant will have enough staff to open?
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04
03
Reliability of a system
Consider the following system with components that are independent of each other. The probability that each individual component works are as follows:
, , , and .
What is the probability that the system works?
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Discrete random variables
05
04
Patients in the hospital
After being discharged from the hospital following a particular surgery, patients often make visits to their local emergency room for treatment. The function below is the CDF of X, the number of emergency room visits per patient:
(a) Find the probability a patient will make more than 1 visit to the emergency room.
(b) Find the probability a patient will not visit the emergency room.
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17
(a)
(b)
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06
02
Gambling with a coin
Two players, A and B, are flipping a fair coin together. If it comes up heads, A pays $1 to B, and if it comes up tails, B pays $1 to A.
They play five rounds. Let
be a random variable recording A’s final winnings. (a) Find the set of possible values of
. (I.e., the set of outcomes with nonzero probability.) (b) Find the PMF and CDF of
. Link to originalSolution
11
(a)
Describe the situation.
Let
be A’s (not total) winnings after the round. Since A either loses one dollar or gains one dollar each round,
.
, so .
(b)
(1) Describe the PMF of
.
if and only if all 5 rounds are tails, so .
if and only if all 5 rounds are heads, so .
if 4 rounds are tails, and if 4 rounds are heads. Since the coin is fair, .
if 3 rounds are heads, and if 3 rounds are tails. Since the coin is fair, .
(2) Define the PMF of
.
(3) Define the CDF of
. Note that the jumps in the PDF are the possible discrete values of
. Link to original