Page 473 - Discrete Mathematics and Its Applications

452 7 / Discrete Probability

35. In roulette, a wheel with 38 numbers is spun. Of these, 18 b) E 1 : the first coin comes up tails; E 2 : two, and not
are red, and 18 are black. The other two numbers, which three, heads come up in a row.
are neither black nor red, are 0 and 00. The probability c) E 1 : the second coin comes up tails; E 2 : two, and not
that when the wheel is spun it lands on any particular three, heads come up in a row.
number is 1/38. (We will study independence of events in more depth in
a) What is the probability that the wheel lands on a red Section 7.2.)
number? 39. Explain what is wrong with the statement that in the
b) What is the probability that the wheel lands on a black Monty Hall Three-Door Puzzle the probability that the
number twice in a row? prize is behind the first door you select and the probabil-
ity that the prize is behind the other of the two doors that
c) What is the probability that the wheel lands on 0 or Monty does not open are both 1/2, because there are two
00?
doors left.
d) What is the probability that in five spins the wheel 40. Suppose that instead of three doors, there are four doors
never lands on either 0 or 00? in the Monty Hall puzzle. What is the probability that you
e) What is the probability that the wheel lands on one of win by not changing once the host, who knows what is
the first six integers on one spin, but does not land on behind each door, opens a losing door and gives you the
any of them on the next spin? chance to change doors? What is the probability that you
36. Which is more likely: rolling a total of 8 when two dice win by changing the door you select to one of the two
are rolled or rolling a total of 8 when three dice are rolled? remaining doors among the three that you did not select?
41. This problem was posed by the Chevalier de Méré and
37. Which is more likely: rolling a total of 9 when two dice was solved by Blaise Pascal and Pierre de Fermat.
are rolled or rolling a total of 9 when three dice are rolled?
a) Find the probability of rolling at least one six when a
38. Two events E 1 and E 2 are called independent if fair die is rolled four times.
p(E 1 ∩ E 2 ) = p(E 1 )p(E 2 ). For each of the following b) Find the probability that a double six comes up at least
pairs of events, which are subsets of the set of all possi- once when a pair of dice is rolled 24 times.Answer the
ble outcomes when a coin is tossed three times, determine query the Chevalier de Méré made to Pascal asking
whether or not they are independent. whether this probability was greater than 1/2.
a) E 1 : tails comes up with the coin is tossed the first c) Is it more likely that a six comes up at least once when
time; E 2 : heads comes up when the coin is tossed the a fair die is rolled four times or that a double six comes
second time. up at least once when a pair of dice is rolled 24 times?

7.2 Probability Theory

Introduction

In Section 7.1 we introduced the notion of the probability of an event. (Recall that an event is a
subset of the possible outcomes of an experiment.) We defined the probability of an event E as
Laplace did, that is,

|E|
p(E) = ,
|S|

the number of outcomes in E divided by the total number of outcomes. This definition assumes
that all outcomes are equally likely. However, many experiments have outcomes that are not
equally likely. For instance, a coin may be biased so that it comes up heads twice as often as
tails. Similarly, the likelihood that the input of a linear search is a particular element in a list, or
is not in the list, depends on how the input is generated. How can we model the likelihood of
events in such situations? In this section we will show how to define probabilities of outcomes
to study probabilities of experiments where outcomes may not be equally likely.
Suppose that a fair coin is flipped four times, and the first time it comes up heads. Given
this information, what is the probability that heads comes up three times? To answer this and