3 Questions for Chapter 3

3.1 Warm-up

Exercise 3.1 Let \(\Omega=\{1,2,3,4,5,6,7,8,9\}\), and assume that all outcomes are equally likely. Let \[\begin{aligned} A&=\{2,4,6,8\}, & B&=\{3,6,9\}, & C&=\{2,3,5,7\}. \end{aligned}\]

Calculate:

\(\mathbb{P}(A \mid B)\);
\(\mathbb{P}(B \mid C)\);
\(\mathbb{P}(C\cup B\mid A)\).

Exercise 3.2 In a survey it is found that 40 percent of the population like dogs, 60 per cent like cats, and 70 per cent of those who like dogs also like cats.

Find the probability that a randomly selected member of the population likes both dogs and cats.
Find the probability that a randomly selected member of the population likes dogs or cats but not both.
What proportion of those who like cats also like dogs?

Exercise 3.3 A multiple-choice question in an exam has 4 possible answers. A student’s knowledge is reflected by the number \(N\) of given answers he is able to eliminate as not being the correct answer (\(N=3\) means he knows the answer; \(N=0\) means he has not got a clue and cannot eliminate any possibilities). Let \(p_k\) be the probability that \(N=k\) (so \(p_0 + p_1 + p_2 +p_3 = 1\)).

Find a formula for the probability that the student gets the answer right, in terms of \(p_0, p_1, p_2, p_3\).
The student gets the answer right. Find a formula for the (conditional) probability that he was certain of the answer beforehand.

Exercise 3.4 Show that if events \(A\) and \(B\) are independent then \(A\) is also independent of \(B^\mathrm{c}\).

hint: Use C1 and C2.

Exercise 3.5 Show that if events \(A\) and \(B\) are independent then the complementary events \(A^\mathrm{c}\), \(B^\mathrm{c}\) are also independent.

hint: Use Exercise 3.4.

Exercise 3.6 Consider a family of two children, whose sexes are represented by the sample space \(\Omega = \{ \text{BB}, \text{BG}, \text{GB}, \text{GG} \}\), with each outcome being equally likely. Consider the events \[B = \{ \text{BB}, \text{BG}, \text{GB} \}, ~ S = \{ \text{BB}, \text{GG} \}, ~ G_1 = \{ \text{GG}, \text{GB} \}, ~\text{and}~ G_2 = \{ \text{GG}, \text{BG} \} .\]

Are \(G_1\) and \(G_2\) independent?
Are \(G_1\) and \(B\) independent?
Show that \(G_1\), \(S\) are independent and that \(G_2\), \(S\) are independent. Are \(G_1, G_2, S\) mutually independent?

Exercise 3.7 A set \(\Omega\) contains \(n\) items. A random subset \(S\) is selected by flipping a fair coin for each item and including it in \(S\) if the coin shows heads.

Use a counting argument to find the number of distinct subsets of \(\Omega\).
Assume that the coins are flipped independently. Use the definition of “independence of multiple events” to show that every subset has an equal chance to be selected.

3.2 Workout

Exercise 3.8 Let \(C\) and \(D\) be any events for which \(\mathbb{P}(C)>0\) and \(\mathbb{P}(D)>0\). Prove that, if \(\mathbb{P}(C \mid D)> \mathbb{P}(C)\), then \(\mathbb{P}(D \mid C) > \mathbb{P}(D)\).

Exercise 3.9 Let \(B \in \mathcal{F}\) be an event with \(\mathbb{P}(B) > 0\). Use the definition of conditional probability to show that \(\mathbb{P}(\, \cdot \mid B)\) satisfies the axioms A1–A4; for example, to verify A3 you must check that for any disjoint events \(C\) and \(D\), \[\mathbb{P}(C \cup D \mid B) = \mathbb{P}(C \mid B) + \mathbb{P}(D \mid B).\]

Exercise 3.10 Suppose that two by-elections are to be held on the same day. Let \(E_1\) denote the event that Labour win seat 1, and \(E_2\) denote the event that Labour win seat 2. Party strategists believe that \(\mathbb{P}(E_1) = 0.6\) and that \(\mathbb{P}(E_2) = 0.4\). Which of the following numbers is the most reasonable estimate for \(\mathbb{P}(E_1\mid E_2)\), and why? \[\text{(a) } 0.3; \quad \text{(b) } 0.4; \quad \text{(c) } 0.5; \quad \text{(d) } 0.6; \quad \text{(e) } 0.7.\]

Exercise 3.11 Consider a family with three children; suppose that boys and girls are equally likely.

What is the probability that there is at least one boy?
What is the (conditional) probability that there is at least one boy, given that there is at least one girl?
What is the (conditional) probability that there is at least one boy, given that the first child is a girl?

Exercise 3.12 You roll two fair dice. What is the conditional probability of rolling at least one six, given that the total score is at least 8?

Exercise 3.13 From a survey of mixed-doubles tennis pairings, it is found that 30% of the men are left-handed, 20% of the women are left-handed, and 50% of the partners of left-handed men are left-handed. A pairing is selected at random from those surveyed.

What is the probability that both of the pair are left-handed?
What is the probability that exactly one of the pair is left-handed?
What is the (conditional) probability that the man is left-handed, given that the woman is left-handed?

Exercise 3.14 Paul is to take a driving test, with probability 0.6 of passing. If he fails he will take a re-test, with a (conditional) probability 0.8 of passing. If he fails the re-test he will take a third test, with a (conditional) probability 0.5 of passing.

What is the probability that Paul fails the first test and passes the second?
What is the probability that he fails all three tests?

Exercise 3.15 Prove property P3 of conditional probability for every \(k\), using P2 and induction.

Exercise 3.16 Re-do the birthday problem (i.e. find the probability that \(n\) people in a room have all different birthdays) using property P3 of conditional probability.

Exercise 3.17 Suppose a proportion 0.001 of the population (i.e., 0.1%) have a certain disease. A diagnostic test is carried out for the disease: the outcome of the test is either positive or negative. It is known from past experience that the test is 90 per cent reliable, i.e. a person with the disease will test (correctly) positive with probability 0.9, while a person without the disease will test (incorrectly) positive with probability 0.1.

A person is tested for the disease. What is the probability that they test positive?
A person’s test result is positive. What is the probability that they have the disease? Comment on your answer.

Exercise 3.18 Consider the following experiment. We have two fair six-sided dice, one red and one blue. A fair coin is tossed; if the coin comes up ‘heads’, the two dice are each rolled twice, while if the coin comes up ‘tails’, the two dice are each rolled just once. All dice rolls are independent, given the number of dice rolls.

Let \(A\) be the event that the total score on the red die is at least 6. Let \(B\) be the event that the total score on the blue die is at least 6.

Are \(A\) and \(B\) independent? Or conditionally independent? Justify your answer.
Given that \(A\) and \(B\) both occur, find the (conditional) probability that the coin came up ‘heads’.

Exercise 3.19 Prove the “equivalent forms for independence” theorem , using the definition of conditional probability.

Exercise 3.20 Show that if \(A\), \(B\), \(C\) are independent then \(A\), \(B\) and \(C^\mathrm{c}\) are independent.

hint: Use C1, C2, and Exercise 3.4.

Exercise 3.21 Three players, A, B, and C, take it in turns (in that order) to roll a standard fair six-sided die. The first player who rolls a 6 wins the game. What are the probabilities of victory for each of the three players?

3.3 Stretch

Exercise 3.22 A traveller arrives at dusk at a small village. She knows that 5% of the villagers are vampires, 10% are werewolves and the remainder are, for want of a better word, normal. She meets a villager, and asks him if he is normal. She knows that normal people always answer this question truthfully. She also knows that vampires are basically honest, so that any particular vampire has only a 10% chance of lying to this question. Werewolves are less honest and have a 30% chance of lying.

What is the probability that the villager will claim to be normal?
The villager claims to be normal. What is the probability that he is lying?

Exercise 3.23 A plane is missing, and it is presumed that it was equally likely to have gone down in any of three possible regions. The probability that the plane will be found upon a search of region 1, when the plane actually is in that region, is judged to be 0.9. The corresponding figures for regions 2 and 3 are 0.95 and 0.85 respectively. What is the conditional probability that the plane is in region 3 given that searches in regions 1 and 2 have been unsuccessful?

Exercise 3.24 Alan, Bernard and Charles are prisoners of the mad king, Max. Max has decided to kill two of the prisoners and release the third, but the prisoners have no information as to who is to be set free. Alan says to the jailer (who knows Max’s decision) “As I know that at least one of my comrades is to be executed, you will give me no information if you tell me the name of one prisoner, other than myself, who is to be executed.”

The jailer accepts this argument, and tells Alan that Bernard is to be executed. Alan now reasons that either he or Charles is to be released, so that his probability of release has increased from 1/3 to 1/2. Which argument is correct, the argument that convinced the jailer or the one that Alan used or neither?

Hint: Let \(A\) denote the event that Alan is to be executed, \(B\) the event that Bernard is to be executed and \(J_B\) the event that the jailer names Bernard. Show that \(B\) is not the same as \(J_B\) by finding \(\cpr{B}{A^\mathrm{c}}\) and \(\cpr{J_B}{A^\mathrm{c}}\).

Exercise 3.25 Suppose a murder has been committed on an island inhabited by \(n+2\) people. You can assume that one of them did it.

A suspect is identified. Genetic material left at the scene of the crime matches that of the suspect. It is not known whether the genetic material of the other inhabitants of the island matches that found at the scene of the crime. The police estimate the chance that the suspect is guilty to be \(\alpha/(n+2)\). All others on the island are a priori assumed equally likely to be guilty.

The proportion of people in the general population of the world, whose genetic characteristics match those of the material at the scene of the crime is \(p\). Matters are complicated by the fact that the suspect has one relative living on the island, and because of this relationship this relative has probability \(q\) of having genetic characteristics that match those of the material at the scene of the crime.

Find a formula for the probability that the suspect is guilty, given the evidence.

Exercise 3.26 At a party you hear that Leo’s birthday is in an earlier month of the year than Matt’s (denote this event by ‘\(L < M\)’). Then you meet Keanu and wonder “what is the chance that Keanu’s birthday is in an earlier month than Leo’s?”. Suppose that each of the three is equally likely to be born in any month of the year and calculate:

\(\pr{L < M}\); (hint: Use P4, with the partition of events \(L_j\) = Leo’s birthday is in month \(j\), with \(j=1,2,\dots, 12\).)
\(\cpr{L_j}{L < M}\), for \(j = 1,2,\dots,12\);
\(\cpr{K < L}{L < M}\).

hint: For the final part of the question, use the conditional version of the partition theorem i.e., \[ \cpr{A}{C} = \sum_j \pr{A \mid B_j\cap C}\cpr{B_j}{C} \] where the \(B_j\) form a partition; also use that conditionally on \(L_j\), \(K < L\) is independent of \(L<M\).

Exercise 3.27 Continuing from Exercise 3.7, generate two subsets \(S_1\) and \(S_2\) independently, and find

\(\pr{|S_1| = m}\) for \(0 \leq m \leq n\), (where \(|S_1|\) is the number of elements of \(S_1\));
\(\pr{S_1\subseteq S_2}\); (hint: use P4)
\(\pr{S_1 \cup S_2 = \Omega}\). (hint: use part b))

You may use that any specific subset \(S\) is generated with probability \(2^{-n}\) (as proven in Exercise 3.7).