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Problem set 2 STAT 2311 FALL 2017

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STAT 2311
FALL 2017
Problem set 2
(due Tuesday, October 17 at the start of your section’s lecture)
1. A system is composed of 4 components, each of which is either working or failed.
Consider an experiment that consists of observing the status of each component, and let
the outcome of the experiment be given by the vector (x1, x2, x3, x4), where xi is equal to
1 if component i is working and is equal to 0 if component i is failed.
a) How many outcomes are in the sample space of this experiment?
b) Suppose that the system will work if components 1 and 2 are both working, or if
components 3 and 4 are both working,. Let W be the event that the system will
work. Specify all the outcomes in W.
c) Let A be the event that components 2 and 4 are both failed. How many
outcomes are contained in the event A?
d) Write out all the outcomes in the event A  W.
2. Two dice are thrown. Let E be the event that the sum of the dice is even, Let F be the
event that at least one of the dice lands on 5, and let G be the event that the sum is 3.
Describe the events E  F, E  F, F  G, E  FC, and E  F  G.
3. In an experiment, a coin is flipped continually until it comes up heads, at which point the
experiment stops. What is the sample space of this experiment? If E8 is the event that
less than 8 flips are required to complete the experiment, how many outcomes are
contained in E8?
4. A 5-person basketball team consists of a point guard, a shooting guard, a forward, a
power forward, and a center.
a) If a person is chosen at random from each of 5 different such teams, what is the
probability of selecting a complete team?
b) What is the probability that all 5 players selected play the same position?
c) What is the probability that 3 players play the same position and the remaining
two play different positions?
5. An urn contains 6 red, 6 blue, and 6 green balls. If a set of 3 balls is randomly selected,
what is the probability that each of the balls will be (a) of the same color? (b) of
different colors? Repeat under the assumption that whenever a ball is selected, its color
is noted and it is then replaced in the urn (i.e. put back in the urn) before the next ball is
selected.
6. If 3 married couples and 2 unmarried persons are arranged in a row, find the probability
that no husband sits next to his wife? (Hint: Use the inclusion-exclusion principle.)
7. An instructor gives her class a set of 30 problems with the information that the final
exam will consist of a random selection of 6 of them. If the student has figured out how
to solve 20 of the problems, what is the probability that he or she will answer correctly
a) All 6 questions on the exam?
b) At least 4 questions?
c) Exactly 3 questions?
8. A 14-hand card is dealt from a well shuffled deck of 52 cards. What is the probability
that the hand contains at least three cards from each of the four suits? What is the
probability that the hand has either 7 spades or 7 diamonds? What is the probability
that the hand has either 5 spades or 2 kings?
9. Seven balls are randomly withdrawn from an urn that contains 5 red, 8 blue, and 7
green balls. Find the probability that
a) 3 red, 2 blue, and 2 green balls are withdrawn;
b) at least 4 red balls are withdrawn;
c) all withdrawn balls are of the same color;
d) either 3 red balls or 3 blue balls or 3 green balls are withdrawn.
10. Suppose 8 teams qualify for the ADA football tournament quarterfinals. The quarterfinal
pairings will be determined by a random draw in which any team can be paired with any
other. The teams are Evils, Back Line, Lyceum United, Undefeated Armada, MBA Eagles,
Front Line, Defeated Armada, Angels. What is the probability that Evils will play Back
Line? What is the probability that Back Line will be paired with Undefeated Armada and
MBA Eagles will be paired with Lyceum United?
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