Chapter 6: Probability

6.1 Introduction

The probability of an event is a number between 0 and 1 that

expresses the long-run likelihood that the event will occur.

An event having probability 0.1 is rather unlikely to occur.

An event with probability 0.9 is very likely to occur.

An event with probability 0.5 is just as likely to occur as not.

Example: A clinic tests for active pulmonary tuberculosis. If a

person has tuberculosis, the probability of a positive test result is

0.98. If a person does not have tuberculosis, the probability of a

negative test result is 0.99. The incidence of tuberculosis in a certain

city is 2 cases per 10,000 population. What is the probability that an

individual who tests positive actually has pulmonary tuberculosis?

6.2 Experiments, Outcomes, and Events

An experiment is an activity with an observable outcome. Each

repetition of the experiment is called a trial. In each trial we

observe the outcome of the experiment.

Experiment 1: Flip a coin

Trial: One coin flip Outcome: Heads

Experiment 2: Allow a conditioned rat to run a maze containing

three possible paths

Trial: One run Outcome: Path 1

Experiment 3: Tabulate the amount of rainfall in Ceres, CA in a year

Trial: One year Outcome: 11.23 in.

The sample space of an experiment is the set of all possible

outcomes of the experiment.

Example: An experiment consists of throwing two dice, one red and

one green, and observing the numbers on the uppermost face on

each. What is the sample space S of this experiment?

An event E is a subset of the sample space.

Example: For the experiment of rolling two dice, describe the

following events:

E1 = {The sum of the numbers is greater than 9};

E2 = {The sum of the numbers is 7 or 11}.

Let S be the sample space of an experiment.

The event corresponding to the empty set, is called the impossible

event, since it can never occur.

The event corresponding to the sample space itself, S, is called the

certain event because the outcome must be in S.

Let E and F be two events of the sample space S.

The event where either E or F or both occurs is designated by E F.

The event where both E and F occurs is designated by E F.

The event where E does not occur is designated by E '.

For the experiment of rolling two dice, let

E1 = “The sum of the numbers is greater than 9” and

E3 = “The numbers on the two dice are equal”.

Determine the sets

E1 E3, E1 E3, and (E1 E3)'.

6.3 Assignment of Probabilities

Suppose you took a coin and tossed it 200 times.

Number Relative frequency

Heads 68 68/200 = 34%

Tails 132 132/200 = 66%

Total 200 1 or 100%

The experimental probability that heads occurs is 34% and

that tails occurs is 66%.

Probability Distribution for the roll of a die

Outcome Probability Outcome Probability 1 1/6 4 1/6 2 1/6 5 1/6 3 1/6 6 1/6

Traffic engineers measure the volume of traffic on a major

highway during the rush hour. Generate a probability distribution

using the data generated over 300 consecutive weekdays.

Assign a probability distribution to this experiment.

Let an experiment have outcomes s1, s2, … , sN with probabilities

p1, p2, … , pN. Then the numbers p1, p2, … , pN must satisfy:

Fundamental Property 1

Each of the numbers p1, p2, … , pN is between 0 and 1;

Fundamental Property 2

p1 + p2 + … + pN = 1.

Addition Principle

Suppose that an event E consists of the finite number of outcomes

s, t, u, … ,z. That is E = {s, t, u, … ,z }.

Then

Pr(E) = Pr(s) + Pr(t) + Pr(u) + … + Pr(z),

Inclusion-Exclusion Principle

Pr( ) Pr( ) Pr( ) Pr( ).E F E F E F

Let E and F be any events. Then

If E and F are mutually exclusive, then

Pr( ) Pr( ) Pr( ).E F E F

Converting between odds and probability

If the odds in favor of an event E are a to b, then

Pr( ) and Pr( ) .a b

E Ea b a b

On average, for every a + b trials, E will occur a times

and E will not occur b times.

6.4 Calculating Probabilities of an Event

Let S be a sample space consisting of N equally likely

outcomes. Let E be any event. Then

number of outcomes in Pr( ) .

EE

N

Complement Rule

Let E be any event, E ' its complement. Then Pr(E) = 1 - Pr(E ').

6.5 Conditional Probability and Independence

Let E and F be events is a sample space S. The conditional

probability, Pr( E | F ) is the probability of event E occurring

given the condition that event F has occurred. In calculating

this probability, the sample space is restricted to F.

provided that Pr(F) ≠ 0.

Pr( )Pr( | )

Pr( )

E FE F

F

Product Rule

If Pr(F) ≠ 0, Pr(E F) = Pr(F) Pr(E | F).

The product rule can be extended to three events.

Pr(E1 E2 E3) = Pr(E1) Pr(E2 | E1) Pr(E3| E1 E2)

Let E and F be events. We say that E and F are independent

provided that Pr(E F) = Pr(E) Pr(F).

Equivalently, they are independent provided that

Pr(E | F) = Pr(E) and Pr(F | E) = Pr(F).

A set of events is said to be independent if, for each collection

of events chosen from them, say E1, E2, …, En, we have

Pr(E1 E2 … En) = Pr(E1) Pr(E2) … Pr(En).