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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
1
2
3
Probability
1/6
1/6
1/6
Outcome
4
5
6
Probability
1/6
1/6
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
Let E and F be any events. Then
Pr( E  F )  Pr( E )  Pr( F )  Pr( E  F ).
If E and F are mutually exclusive, then
Pr( E  F )  Pr( E )  Pr( F ).
Converting between odds and probability
If the odds in favor of an event E are a to b, then
a
b
Pr( E ) 
and Pr( E ) 
.
a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
Pr(E ) 
 number of outcomes in E  .
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.
Pr( E  F )
Pr( E | F ) 
Pr( F )
provided that Pr(F) ≠ 0.
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).