A fair coin is tossed repeatedly. Let s denote any fixed sequence of heads and tails of length r . Showthat with probability one the sequence s will eventually appear in r con-secutive tosses of the coin.
(The usual statement of Murphy’s law says that anything that can go wrong, will go wrong).
Solution If a fair coin is tossed r times, there are 2r distinct equally likely outcomes and one of them is s. We consider a fair die with 2r faces; each face corresponds to one
Problems 47 of the 2r outcomes of tossing the coin r times and one of them is face s. Nowroll the die repeatedly.
Let Ak be the event that face s appears for the first time on the kth roll. There are 2r k distinct outcomes of k rolls, and by symmetry they are equally likely. In (2r − 1)k−1 of them, Akoccurs, so by (1.3.1),
which is the probability that face s appears at all in m rolls.
Nowconsider n tosses of the coin, and let m= [nr] (where [x] is the integer part of x).
The n tosses can thus be divided into m sequences of length r with a remainder n− mr.
Let Bn be the event that none of these m sequences is s, and let Cn be the event that the sequence s does not occur anywhere in the n tosses. Then
Cn ⊆ Bn =
because rolling the die m times and tossing the coin mr times yield the same sample space of equally likely outcomes. Hence, by Example 1.4.11(i) and (1.4.5) and (1),
P(Cn)≤ P(Bn)=
2r− 1 2r
m
→ 0 as n→ ∞. Nowthe event that s eventually occurs is lim
n→∞(Cnc), so by (1.4.5) and (1.5.4),
(2) Exercise If the coin is tossed n times, showthat the probability that it shows heads on an odd number of tosses (and tails on the rest) is12.
(3) Exercise If the coin is tossed an unbounded number of times, showthat the probability that a head is first shown on an odd numbered toss is23.
(4) Exercise If Malone tosses his coin m times and Watt tosses his coin n times, showthat the probability that they get the same number of heads each is equal to the probability that Beckett gets m heads in m+ n tosses of his coin.
P R O B L E M S
N.B. Unless otherwise stated, coins are fair, dice are regular cubes and packs of cards are well shuffled with four suits of 13 cards.
1 You are given a conventional pack of cards. What is the probability that the top card is an ace?
2 You count a pack of cards (face down) and find it defective (having only 49 cards!). What is the probability that the top card is an ace?
3 A class contains seven boys and eight girls.
(a) If two are selected at random to leave the room, what is the probability that they are of different sexes?
(b) On two separate occasions, a child is selected at random to leave the room. What is the probability that the two choices result in children of different sexes?
4 An urn contains 100 balls numbered from 1 to 100. Four are removed at random without being replaced. Find the probability that the number on the last ball is smaller than the number on the first ball.
5 LetF be an event space. Showthat the total number of events in F cannot be exactly six. What integers can be the number of events in a finite event space?
6 To start playing a game of chance with a die, it is necessary first to throw a six.
(a) What is the probability that you throwyour first six at your third attempt?
(b) What is the probability that you require more than three attempts?
(c) What is the most likely number of attempts until you first throwa six?
(d) After how many throws would your probability of having thrown a six be at least 0.95?
7 Let A, B, and C be events. Write down expressions for the events where (a) At least two of A, B, and C occur.
(b) Exactly two of A, B, and C occur.
(c) At most two of A, B, and C occur.
(d) Exactly one of A, B, and C occurs.
8 A die is loaded in such a way that the probability that a 6 is thrown is five times that of any other number, each of them being equally probable.
(a) By what factor is the probability of a total of 24 from four throws greater than that for an unloaded die?
(b) Showthat for the loaded die, the probability of obtaining a total of six from four throws is two and half times that of obtaining five, and compare the probability of obtaining 23 with that of obtaining 24 from four throws.
9 A fair coin is tossed four times. What is the probability of (a) At least three heads?
(b) Exactly three heads?
(c) A run of three or more consecutive heads?
(d) A run of exactly three consecutive heads?
10 Find the probability that in 24 throws of two dice, double six fails to appear.
11 Two dice are rolled and their scores are denoted by S1 and S2. What is the probability that the quadratic x2+ x S1+ S2= 0 has real roots?
12 (a) If P( A) is the probability that an event A occurs, prove that
P
(b) A tea set consists of six cups and saucers with two cups and saucers in each of three different colours. The cups are placed randomly on the saucers. What is the probability that no cup is on a saucer of the same colour?
13 An urn contains three tickets numbered 1, 2, and 3, and they are drawn successively without replacement. What is the probability that there will be at least one value of r (r= 1, 2, 3) such that on the r th drawing a ticket numbered r will be drawn?
Problems 49 14 Four red balls and two blue balls are placed at random into two urns so that each urn contains three
balls. What is the probability of getting a blue ball if (a) You select a ball at random from the first urn?
(b) You select an urn at random and then select a ball from it at random?
(c) You discard two balls from the second urn and select the last ball?
15 Four fair dice are rolled and the four numbers shown are multiplied together. What is the probability that this product
(a) Is divisible by 5?
(b) Has last digit 5?
16 Suppose that n fair dice are rolled, and let Mnbe the product of the numbers shown.
(a) Showthat the probability that the last digit of Mnis 5 is a nonincreasing function of n.
(b) Showthat the probability that Mnis divisible by 5 is a non-decreasing function of n.
(c) Find the limits of the probabilities in (a) and (b) and interpret this.
17 The consecutive integers 1, 2, . . . , n are inscribed on n balls in an urn. Let Drbe the event that the number on a ball drawn at random is divisible by r .
(a) What are P(D3), P(D4), P(D3∪ D4), and P(D3∩ D4)?
(b) Find the limits of these probabilities as n→ ∞.
(c) What would your answers be if the n consecutive numbers began at a number a= 1?
18 Showthat if A and B are events, then
P( A∩ B) − P(A)P(B) = P(A)P(Bc)− P(A ∩ Bc)
= P(Ac)P(B)− P(Ac∩ B)
= P((A ∪ B)c)− P(Ac)P(Bc) 19 Showthat
(a) min{1, P(A) + P(B)} ≥ P(A ∪ B) ≥ max {P(A), P(B)} . (b) min{P(A), P(B)} ≥ P(A ∩ B) ≥ max {0, P(A) + P(B) − 1} . (c) P
n
1
Ai
≥n
i=1P( Ai)− (n − 1).
20 The function d(x, y) is defined on the event space by d(A, B) = P(A B).
(a) Showthat for any events A, B, and C,
d( A, B) + d(B, C) − d(A, C) = 2(P(A ∩ Bc∩ C) + P(Ac∩ B ∩ Cc)). (b) When is d( A, B) zero?
(c) Let A1, A2, . . . be a monotone sequence of events such that Ai⊆ Aj for i≤ j. Showthat for i≤ j ≤ k,
d( Ai, Ak)= d(Ai, Aj)+ d(Aj, Ak).
21 An urn contains x≥ 2 xanthic balls and y ≥ 1 yellow balls. Two balls are drawn at random without replacement; let p be the probability that both are xanthic.
(a) If p= 12, find the smallest possible value of x in the two cases when y is odd or even.
(b) If p= 18, find the smallest possible value of x.
(c) If p= r−2, where r is an integer, showthat r ≥ 6, and find values of x and y that yield p =361. 22 When are the following true?
(a) A∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (b) A∩ (B ∩ C) = (A ∩ B) ∩ C (c) A∪ (B ∪ C) = A\(B\C) (d) ( A\B)\C = A\(B\C)
(e) A (B C) = (A B) C (f) A\(B ∩ C) = (A\B) ∪ (A\C) (g) A\(B ∪ C) = (A\B) ∩ (A\C).
23 Birthdays If m students born in 1985 are attending a lecture, showthat the probability that at least two of them share a birthday is
p= 1 − (365)!
(365− m)!(365)m
Showthat if m≥ 23, then p > 12. What difference would it make if they were born in 1988?
24 Let ( An; n> 1) be a collection of events. Showthat the event that infinitely many of the Anoccur is given by
n≥1
∞ m=nAm. 25Boole’s Inequality Showthat
P n
1
Ai
≤
n i=1
P( Ai).