Tutorial Exercise (Week 7)

1.
MATH2421 T1C
Tutorial Exercise (Week 7)
(Chapter 4 Problem 49) When coin A and coin B are flipped, they land on heads with
probabilities 0.4 and 0.7 respectively. One coin is randomly chosen and flipped 10 times.
Given that the first flip lands on heads, what is the probability that exactly 7 of the 10 flips
land on heads?
2.
(Chapter 4 Problem 53) Estimate the probability that, for at least 1 of 80,000 couples, both
partners celebrate their birthdays on the same day of a year.
3.
(Chapter 4 Problem 60) The number of times that a person catches a cold in a year is a Poisson
random variable with mean 5. Suppose that a new drug that, reduces the Poisson mean to 3 for
75% of the population and, has no effect for the other 25% of the population, has just been
marketed. If a person takes the drug for a year and has 2 colds at that time, what is the
probability that the drug is beneficial for him or her?
4.
(Chapter 4 Problem 63) The number of people entering a casino in a 2-minute interval is a
Poisson random variable with mean 1. What is the probability that no one enters the casino
between 12:00 and 12:05?
5.
(Chapter 4 Problem 65) Each of 500 soldiers in an army independently has a particular disease
with probability 0.001. The presence of this disease can be tested by a blood test. The blood
samples from all 500 soldiers are pooled and tested.
(a) Estimate the probability that the blood test shows a positive result, i.e. the probability that
at least 1 soldier has the disease.
(b) Given that the blood test shows a positive result, estimate the probability that more than 1
soldier has the disease.
(c) 1 of the 500 soldiers is Jones, who knows that he has the disease. In his view, estimate the
probability that more than 1 soldier has the disease.
6.
(Chapter 4 Problem 66) 2n people, consisting of n couples, sit randomly around a round table.
Estimate the probability for large n that no couples sit next to each other.
7.
(Chapter 4 Problem 70) At time 0, a coin that comes up heads with probability p is flipped and
falls to the ground. Suppose it lands on heads. The coin is flipped and falls to the ground at
times chosen according to a Poisson process with rate λ. What is the probability that the coin
shows heads at time t?
8.
(Chapter 4 Problem 72) A and B play a series of games without draws. The first one to win 4
games is declared the overall winner. Suppose that A independently wins each game with
probability 0.6.
(a) Find the probabilities, for i = 4, 5, 6 and 7, that A wins the series in exactly i games.
(b) Find the probabilities that A and B win the series respectively.
9.
(Chapter 4 Problem 75) A fair coin is continually flipped until the 10th head appears. Let X
denote the number of tails that occur. Find the probability mass function of X.
10. (Chapter 4 Problem 76, 77)
(a) At all times, a pipe-smoking mathematician carries 2 matchboxes, 1 in his left-hand pocket
and 1 in his right-hand pocket. Each time he needs a match, he is equally likely to take it
from either pocket. Consider the moment when the mathematician first discovers that one
of his matchboxes is empty. If it is assumed that both matchboxes initially contained N
matches, what is the probability that there are exactly k matches, k = 0, 1, …, N, in the
other box?
(b) Find the probability that, at the moment when the first box is emptied (as opposed to being
found empty), the other box contains exactly k matches.
(c) Solve the problem when the left-hand matchbox originally contained N1 matches and the
right-hand box contained N2 matches.
11. (Chapter 4 Problem 78) An urn contains 4 white and 4 black balls. We randomly choose 4 balls.
If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again
randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the
probability that we shall make exactly n selections?
12. (Chapter 4 Problem 79) Suppose that a batch of 100 items contains 6 that are defective and 94
that are not defective. If X is the number of defective items in a randomly drawn sample of 10
items from the batch, find P(X = 0) and P(X > 2).
13. (Chapter 4 Problem 80) A game is played as follows: Twenty numbers are selected at random
by the casino from the set of numbers 1 through 80. A player can select from 1 to 15 numbers;
a win occurs if some fraction of the player’s chosen subset matches any of the 20 numbers
drawn by the house. The payoff is a function of the number of elements in the player’s
selection and the number of matches. Let Pn, k denote the probability that exactly k of the n
numbers chosen by the player are among the 20 selected by the house. Find Pn, k.
14. (Chapter 4 Problem 81) A purchaser of electrical components buys them in lots of size 10. It is
his policy to inspect 3 components randomly from a lot and to accept the lot only if all 3 are
nondefective. 30 percent of the lots have 4 defective components and 70 percent have only 1.
Given that a lot is rejected, what is the probability that it contained 4 defective components?
15. (Chapter 4 Problem 82) A purchaser of transistors buys them in lots of 20. It is his policy to
randomly inspect 4 components from a lot and to accept the lot only if all 4 are nondefective.
If each component in a lot is independently defective with probability 0.1, what proportion of
lots is rejected?