If 85% of people have a bowl of cereal for breakfast, 60% of people have some toast for
breakfast, and 50% of people have both cereal and toast for breakfast, what proportion of
people have neither cereal nor toast for breakfast?
If 50 percent of families in a certain city subscribe to the morning newspaper, 65 percent
subscribe to the afternoon newspaper, and 85 percent of the families subscribe to at least one
of the two newspapers, what proportion of families subscribe to both newspapers?
(a) A student is to answer 7 out of 10 questions in an examination. How many choices has
she? How many choices does she have if at least 3 of the first 5 questions have to be
answered? Hint: Split the exam into the first 5 and last 5 questions.
(b) A line-up of 10 men is conducted in order that a witness can identify 3 suspects. Suppose
that 3 people in the line-up actually committed the crime in question. If the witness
does not recognise any of the suspects, but simply chooses three men at random, what
is the probability that the three guilty men are selected? What is the probability that the
witness selects three innocent men?
(a) What is the probability of being dealt two Aces and two Kings in a hand of Poker? In
Poker, you are dealt 5 cards from a pack of 52. The 52 cards are made up of 13 cards
in each of 4 suits. The “other” card should not be an Ace or a King, as this would give
a Full House.
(b) What is the probability of being dealt Two Pairs in a hand of Poker. Again, be sure not
to include Full Houses.
(a) What is the probability of getting exactly 4 numbers on the National Lottery?
(b) What is the probability of getting the Jackpot (all 6 main numbers)?
(c) What is the probability of getting 5 main numbers plus the bonus ball?
(d) What is the probability of winning a prize (getting at least 3 numbers)?
(e) If you play one line per week for 2080 weeks (40 years), what is the probability that
you win the Jackpot at least once?
Capture-recapture is a widely used statistical technique in environmental statistics.
(a) A lake contains N fish. One day n of these are caught, marked with a tag, and returned
to the lake. A week later, m fish are caught. What is the probability that k of those m
are marked?
(b) If the lake contains 1,000 fish, 100 of which are marked, and you catch 10, what is the
probability that at least one of the fish you catch is marked?
14. In a squash tournament between three players A;B and C, each player plays the others once
(ie. A plays B, A plays C and B plays C). Assume the following probabilities:
P(A beats B) = 0:6; P(A beats C) = 0:7; P(B beats C) = 0:6:
Assuming independence of the match results, calculate the probability that A wins at least
as many games as any other player.
15. An insurance company classifies drivers as class X, Y or Z. Experience indicates that the
probability a class X driver has at least one accident in any one year is 0.02, while the
corresponding probabilities for Y and Z are 0.04 and 0.1 respectively. They have found
that of the drivers who apply to them for cover, 30% are class X, 60% are class Y , and
10% are class Z. Assume that for each class of driver, accidents in subsequent years occur
independently. However, for a driver of unknown class, accidents in subsequent years will
not occur independently.
(a) What the the probability of a new client having an accident in their first year?
(b) What is the probability of a new client having no accidents in their first 5 years?
(c) If a new client has no accidents in their first five years, what is the probability that they
are a class X driver.
16. A player rolls a die. They stop at the fourth roll or when a six appears, whichever occurs
first. Let X be the number of rolls. Find the PMF for X, and hence calculate E(X) and
Var (X).
17. A medical experiment consists of subjecting a patient to a certain stimulus and recording
whether or not they respond. The probability of a response is p. A sequence of 5 independent
trials is called a series.
(a) What is the probability that there is exactly one response in a series?
(b) What is the probability that there is at least one response in a series?
(c) What is the probability that, in 8 independent series, exactly two consist entirely of
responses.
18. Whether or not certain mice are black or brown is genetically determined by a pair of alleles,
each of which is either B or b. If both alleles are alike, then the mouse is said to be homozygous
in that gene, and if they are different, the mouse is said to be heterozygous. So, for this
gene, BB and bb are homozygous, and Bb and bB are heterozygous. A mouse is only brown
if it is homozygous bb. If it is BB, Bb or bB, then it is black. Here, the brown allele b is said
to be recessive, and the black allele, B, is said to be dominant.
The offspring of a pair of mice receive one allele from each parent, and the allele that they
get is randomly chosen from the two possible with equal probability. So, if a parent is heterozygous,
it is equally likely to donate a B or a b allele to its offspring. If it is homozygous,
the allele donated to the offspring is determined. Thus, a brown mouse, bb is certain to donate
a b allele to its offspring. Whether or not the offspring is brown will then depend on the
allele donated by the other parent.
For this question, suppose that two heterozygous mice are mated.
(a) What is the probability that a homozygous BB mouse results?
(b) What is the probability that a black offspring results?
In fact, when these two heterozygous mice are mated, a black mouse results. Call this black
mouse “Mickey”.
(c) What is the probability that Mickey is homozygous BB?
(d) If Mickey is mated with a brown mouse, what is the probability that its offspring will
be brown?
(e) If Mickey is mated with a heterozygous mouse, what is the probability that its offspring
will be brown?
Now suppose that Mickey is mated with a brown mouse, resulting in seven offspring, all of
which turn out to be black.
(f) Conditional on this information, what is the probability that Mickey is in fact homozygous
BB?
21. Suppose that the number of times during a year that an individual catches a cold can be
modelled by a Poisson random variable with an expectation of 4. Further suppose that a new
drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for
80% of the population, but has no effect on the remaining 20% of the population. Calculate
(a) the probability that an individual taking the drug has 2 colds in a year if they are part
of the population which benefits from the drug;
(b) the probability that an individual has 2 colds in a year if they are part of the population
which does not benefit from the drug;
(c) the probability that a randomly chosen individual has 2 colds in a year if they take the
drug;
(d) the conditional probability that a randomly chosen individual is in that part of the population
which benefits from the drug given that they had 2 colds in a year during which
they took the drug.
22. Each day a hospital makes available two beds to each of two surgeons (four beds in total).
The demand each surgeon has for these beds is assumed to be independently Poisson with
expectation 1. Calculate the probability
(a) that the demand for beds for a particular surgeon exceeds those available;
(b) that the demand for beds for at least one of the two surgeons exceeds the number
available;
(c) that the demand exceeds the number of available beds if the two surgeons decide to cooperate
and pool their resources. Hint: remember that the sum of independent Poissons
is Poisson.
breakfast, and 50% of people have both cereal and toast for breakfast, what proportion of
people have neither cereal nor toast for breakfast?
If 50 percent of families in a certain city subscribe to the morning newspaper, 65 percent
subscribe to the afternoon newspaper, and 85 percent of the families subscribe to at least one
of the two newspapers, what proportion of families subscribe to both newspapers?
(a) A student is to answer 7 out of 10 questions in an examination. How many choices has
she? How many choices does she have if at least 3 of the first 5 questions have to be
answered? Hint: Split the exam into the first 5 and last 5 questions.
(b) A line-up of 10 men is conducted in order that a witness can identify 3 suspects. Suppose
that 3 people in the line-up actually committed the crime in question. If the witness
does not recognise any of the suspects, but simply chooses three men at random, what
is the probability that the three guilty men are selected? What is the probability that the
witness selects three innocent men?
(a) What is the probability of being dealt two Aces and two Kings in a hand of Poker? In
Poker, you are dealt 5 cards from a pack of 52. The 52 cards are made up of 13 cards
in each of 4 suits. The “other” card should not be an Ace or a King, as this would give
a Full House.
(b) What is the probability of being dealt Two Pairs in a hand of Poker. Again, be sure not
to include Full Houses.
(a) What is the probability of getting exactly 4 numbers on the National Lottery?
(b) What is the probability of getting the Jackpot (all 6 main numbers)?
(c) What is the probability of getting 5 main numbers plus the bonus ball?
(d) What is the probability of winning a prize (getting at least 3 numbers)?
(e) If you play one line per week for 2080 weeks (40 years), what is the probability that
you win the Jackpot at least once?
Capture-recapture is a widely used statistical technique in environmental statistics.
(a) A lake contains N fish. One day n of these are caught, marked with a tag, and returned
to the lake. A week later, m fish are caught. What is the probability that k of those m
are marked?
(b) If the lake contains 1,000 fish, 100 of which are marked, and you catch 10, what is the
probability that at least one of the fish you catch is marked?
14. In a squash tournament between three players A;B and C, each player plays the others once
(ie. A plays B, A plays C and B plays C). Assume the following probabilities:
P(A beats B) = 0:6; P(A beats C) = 0:7; P(B beats C) = 0:6:
Assuming independence of the match results, calculate the probability that A wins at least
as many games as any other player.
15. An insurance company classifies drivers as class X, Y or Z. Experience indicates that the
probability a class X driver has at least one accident in any one year is 0.02, while the
corresponding probabilities for Y and Z are 0.04 and 0.1 respectively. They have found
that of the drivers who apply to them for cover, 30% are class X, 60% are class Y , and
10% are class Z. Assume that for each class of driver, accidents in subsequent years occur
independently. However, for a driver of unknown class, accidents in subsequent years will
not occur independently.
(a) What the the probability of a new client having an accident in their first year?
(b) What is the probability of a new client having no accidents in their first 5 years?
(c) If a new client has no accidents in their first five years, what is the probability that they
are a class X driver.
16. A player rolls a die. They stop at the fourth roll or when a six appears, whichever occurs
first. Let X be the number of rolls. Find the PMF for X, and hence calculate E(X) and
Var (X).
17. A medical experiment consists of subjecting a patient to a certain stimulus and recording
whether or not they respond. The probability of a response is p. A sequence of 5 independent
trials is called a series.
(a) What is the probability that there is exactly one response in a series?
(b) What is the probability that there is at least one response in a series?
(c) What is the probability that, in 8 independent series, exactly two consist entirely of
responses.
18. Whether or not certain mice are black or brown is genetically determined by a pair of alleles,
each of which is either B or b. If both alleles are alike, then the mouse is said to be homozygous
in that gene, and if they are different, the mouse is said to be heterozygous. So, for this
gene, BB and bb are homozygous, and Bb and bB are heterozygous. A mouse is only brown
if it is homozygous bb. If it is BB, Bb or bB, then it is black. Here, the brown allele b is said
to be recessive, and the black allele, B, is said to be dominant.
The offspring of a pair of mice receive one allele from each parent, and the allele that they
get is randomly chosen from the two possible with equal probability. So, if a parent is heterozygous,
it is equally likely to donate a B or a b allele to its offspring. If it is homozygous,
the allele donated to the offspring is determined. Thus, a brown mouse, bb is certain to donate
a b allele to its offspring. Whether or not the offspring is brown will then depend on the
allele donated by the other parent.
For this question, suppose that two heterozygous mice are mated.
(a) What is the probability that a homozygous BB mouse results?
(b) What is the probability that a black offspring results?
In fact, when these two heterozygous mice are mated, a black mouse results. Call this black
mouse “Mickey”.
(c) What is the probability that Mickey is homozygous BB?
(d) If Mickey is mated with a brown mouse, what is the probability that its offspring will
be brown?
(e) If Mickey is mated with a heterozygous mouse, what is the probability that its offspring
will be brown?
Now suppose that Mickey is mated with a brown mouse, resulting in seven offspring, all of
which turn out to be black.
(f) Conditional on this information, what is the probability that Mickey is in fact homozygous
BB?
21. Suppose that the number of times during a year that an individual catches a cold can be
modelled by a Poisson random variable with an expectation of 4. Further suppose that a new
drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for
80% of the population, but has no effect on the remaining 20% of the population. Calculate
(a) the probability that an individual taking the drug has 2 colds in a year if they are part
of the population which benefits from the drug;
(b) the probability that an individual has 2 colds in a year if they are part of the population
which does not benefit from the drug;
(c) the probability that a randomly chosen individual has 2 colds in a year if they take the
drug;
(d) the conditional probability that a randomly chosen individual is in that part of the population
which benefits from the drug given that they had 2 colds in a year during which
they took the drug.
22. Each day a hospital makes available two beds to each of two surgeons (four beds in total).
The demand each surgeon has for these beds is assumed to be independently Poisson with
expectation 1. Calculate the probability
(a) that the demand for beds for a particular surgeon exceeds those available;
(b) that the demand for beds for at least one of the two surgeons exceeds the number
available;
(c) that the demand exceeds the number of available beds if the two surgeons decide to cooperate
and pool their resources. Hint: remember that the sum of independent Poissons
is Poisson.
