Term 2, 22/23
STAT201 PROBABILITY THEORY AND APPLICATIONS
Homework 1
1.
An urn contains 30 chips: 10 white chips numbered 1 through 10, 10 black chips
numbered 1 through 10, and 10 red chips number 1 through 10.
(a) If two chips are to be drawn at random with replacement, what is the probability
that both chips are the same?
(b) If two chips are to be drawn at random with replacement, what is the probability
that both chips will be of same color or bear the same number, but they are not
the same?
(c) If two chips are to be drawn at random without replacement, what is the probability that both chips will be of same color or bear the same number?
2.
A male customer visiting the suit department of a certain store will purchase a suit
with probability 0.16, a shirt with probability 0.18, and a tie with probability 0.2.
The customer will purchase both a suit and a shirt with probability 0.07, both a shirt
and a tie with probability 0.09, and with no probability of purchasing a suit and a
tie.
(a) What is the probability that a customer purchases at least one of these items?
(b) Given that the customer purchases a shirt, what is the probability that he also
purchases a tie?
(c) Given that the customer purchases at least one item, what is the probability
that he purchases a shirt and a tie?
(d) If there are four customers come to visit the suit department in one particular
hour, what is the probability that none of them purchase any items?
3.
A cookie jar has 3 red marbles and 1 black marble. A shoe box has 1 red marble and
1 white marble. An urn has two red marbles and 1 black marble. Three marbles are
chosen at random without replacement from the cookie jar and placed in the shoe
box. Then 2 marbles are chosen at random without replacement from the shoe box
and placed in the urn.
If 3 marbles are chosen at random without replacement from the urn, what is the
probability that 3 marbles have different color?
-PTO-
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4.
A diagnostic test for a certain disease is with sensitivity(true positive rate) 95%,
meaning that, if a person has the disease, the test will show positive with probability
0.95. Also, the diagnostic test is with specificity(true negative rate) 98%, meaning
that if a person does not have the disease, the test will show negative with probability
0.98. Suppose the sensitivity and specificity of the test don’t depend on the gender of
the patients, and there are 1% of male population and 4% of female population have
the disease in question. If the diagnostic test reports that a married couple chosen at
random from the population have the disease, what is the probability that at least
one of them, in fact, has the disease?
5.
Nine students, five men and four women, interview for four summer internships
sponsored by a city newspaper.
(a) In how many ways can the newspaper choose a set of four interns?
(b) In how many ways can the newspaper choose a set of four interns if it must
include two men and two women in each set?
(c) How many sets of four can be picked such that not everyone in a set is of the
same sex?
6.
Four Chinese (C1 , C2 , C3 , C4 ), three Malay (M1 , M2 , M3 ) and three Indians (I1 , I2 , I3 )
are lined up at the box office, waiting to buy tickets for the Singapore Formula One
Race.
(a) How many ways can they positions themselves if the Malay are to hold the first
three places in line; the Chinese, the next four; and the Indians, the last three?
(b) How many arrangements are possible if members of the same race must stay
together?
(c) How many ways can they positions themselves if one of four Chinese is to hold
the first place in line and one of Indians is to hold the last place?
(d) How many different queues can be formed?
7.
In how many ways can the letters in TENNESSEE be arranged so that no two E’s
are adjacent?
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