Chapter 7 Probability

Section 7.1 Experiments, Sample Spaces, and Events

Terminology

Experiment

  • An experiment is an activity with observable results.
  • The results of an experiment are called outcomes of the experiment.

Examples –

Tossing a coin and observing whether it falls heads or tails

Rolling a die and observing which of the numbers 1, 2, 3, 4, 5, or 6 shows up

Testing a spark plug from a batch of 100 spark plugs and observing whether or not it is defective

Sample Point, Sample Space, and Event

  • Sample point:

An outcome of an experiment

  • Sample space:

The set consisting of all possible sample points of an experiment

  • Event:

A subset of a sample space of an experiment

Example:Describe the sample space associated with the experiment of tossing a coin and observing whether it falls heads or tails.What are the events of this experiment?

Union of Two Events

  • The union of two events E and F is the event E F.
  • Thus, the event E  F contains the set of outcomes of E and/or F.

Intersection of Two Events

  • The intersection of two events E and F is the event E F.
  • Thus, the event E  F contains the set of outcomes common to E and F.

Complement of an Event

  • The complement of an event E is the event E c.
  • Thus, the event E c is the set containing all the outcomes in the sample space S that are not in E.

Example:Consider the experiment of rolling a die and observing the number that falls uppermost.Let

S = {1, 2, 3, 4, 5, 6} denote the sample space of the experiment and E = {2, 4, 6} and F = {1, 3} be events of this experiment.Compute E F. Interpret your results.Compute E F. Interpret your results. Compute

F c. Interpret your results

Mutually Exclusive Events

  • E and F are mutually exclusive if E F = Ø

Example: An experiment consists of tossing a coin three times and observing the resulting sequence of heads and tails.Describe the sample space S of the experiment.Determine the event E that exactly two heads appear.Determine the event F that at least one head appears.

Example:Applied Example: Movie Attendance

The manager of a local cinema records the number of patrons attending a first-run movie screening.The theatre has a seating capacity of 500.Determine an appropriate sample space for this experiment.Describe the event E that fewer than 50 people attend the screening. Describe the event F that the theatre is more than half full at the screening.

Section 7.2Definition of Probability

Probability of an Event in a Uniform Sample Space

If S = {s1, s2, … , sn} is the sample space for an experiment in which the outcomes are equally likely, then we assign the probabilities to each of the outcomes s1,s2,… , sn.

Example: Tossing a Coin

If a single fair coin is tossed, find the probability that it will land heads up.

Theoretical Probability Formula

If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by

Example:A fair die is rolled, and the number that falls uppermost is observed. Determine the probability distribution for the experiment.

Finding the Probability of Event E

  1. Determine a sample space S associated with the experiment.
  2. Assign probabilities to the simple events of S.
  3. If E = {s1, s2, … , sn} where {s1}, {s2}, {s3}, … , {sn} are simple events, then

P(E) = P(s1) + P(s2) + P(s3) + ··· + P(sn) If E is the empty set, Ø, then P(E) = 0.

Example: Flipping a Cup

A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top.

Empirical Probability Formula

If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by

Example: Card Hands

There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush.

Example: Gender of a Student

A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female?

Section 7.3 Rules of Probability

Properties of the Probability Function

Property 1.P(E)  0 for every E.

Property 2.P(S) = 1.

Property 3.If E and F are mutually exclusive (E  F = Ø), then

P(E  F) = P(E) + P(F)

Example: Applied Example: SAT Verbal Scores

The superintendent of a metropolitan school district has estimated the probabilities associated with the SAT verbal scores of students from that district.The results are shown in the table below.

If a student is selected at random, find the probability that his or her SAT verbal score will be

More than 400.

Less than or equal to 500.

Greater than 400 but less than or equal to 600.

Score, x / Probability
x > 700 / .01
600 < x  700 / .07
500 < x  600 / .19
400 < x  500 / .23
300 < x  400 / .31
x  300 / .19

Addition Rule

Property 4.If E and F are any two events of an experiment, then

P(E  F) = P(E) + P(F) – P(E  F)

Example:A card is drawn from a shuffled deck of 52 playing cards.What is the probability that it is an ace or a spade?

Example: Applied Example: Quality Control

The quality-control department of Vista Vision, manufacturer of the Pulsar plasma TV, has determined from records obtained from the company’s service centers that 3% of the sets sold experience video problems, 1% experience audio problems, and 0.1% experience both video and audio problems before the expiration of the warranty.Find the probability that a plasma TV purchased by a consumer will experience video or audio problems before the warranty expires.

Rule of Complements

Property 5.If E is an event of an experiment and Ec denotes the complement of E, then

P(Ec) = 1 – P(E)

Example: Applied Example: Warranties

What is the probability that a Pulsar plasma TV (from the last example) bought by a consumer will not experience video or audio problems before the warranty expires?

Section 7.4 Use of Counting Techniques in Probability

Computing the Probability of an Event in a Uniform Sample Space

  • Let S be a uniform sample space and let E be any event. Then,

Events Involving “Not” and “Or”

Properties of Probability

Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold.

Example: Rolling a Die

When a single fair die is rolled, find the probability of each event.

a) the number 3 is rolled

b) a number other than 3 is rolled

c) the number 7 is rolled

d) a number less than 7 is rolled

Events Involving “Not”

The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction.

Correspondences

Set Theory / Logic / Arithmetic
Operation or Connective (Symbol) / Complement / Not / Subtraction
Operation or Connective (Symbol) / Union / Or / Addition
Operation or Connective (Symbol) / Intersection / And / Multiplication

Probability of a Complement

The probability that an event E will not occur is equal to one minus the probability that it will occur.

Example: Complement

When a single card is drawn from a standard 52-card deck, what is the probability that is will not be an ace?

Events Involving “Or”

Probability of one event or another should involve the union and addition.

Mutually Exclusive Events

Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.)

Addition Rule of Probability (for A or B)

If A and B are any two events, then

If A and B are mutually exclusive, then

Example: Probability Involving “Or”

When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or a diamond?

Example: Probability Involving “Or”

If a single die is rolled, what is the probability of a 2 or odd?

Section 7.5 Conditional Probability and Independent Events

Conditional Probability of an Event

  • If A and B are events in an experiment and P(A) ¹ 0, then the conditional probability that the event B will occur given that the event A has already occurred is

Conditional Probability

Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen – the timing is not important). This type of probability is called conditional probability.

The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B, given A, and is denoted P(B | A).

Example: Selecting From a Set of Numbers

From the sample space S = {2, 3, 4, 5, 6, 7, 8, 9}, a single number is to be selected randomly. Given the events

A: selected number is odd, and

B selected number is a multiple of 3.

find each probability.

a) P(B)

b) P(A and B)

c) P(B | A)

Conditional Probability Formula

The conditional probability of B, given A, and is given by

Example: Probability in a Family

Given a family with two children, find the probability that both are boys, given that at least one is a boy.

Independent Events

Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if

P(B | A) = P(B), or equivalently

P(A | B) = P(A).

Example: Checking for Independence

A single card is to be drawn from a standard 52-card deck. Given the events

A: the selected card is an ace

B: the selected card is red

a) Find P(B).

b) Find P(B | A).

c) Determine whether events A and B are independent.

Events Involving “And”

If we multiply both sides of the conditional probability formula by P(A), we obtain an expression for P(A and B). The calculation of P(A and B) is simpler when A and B are independent.

Multiplication Rule of Probability

If A and B are any two events, then

If A and B are independent, then

Example: Selecting From an Jar of Balls

Jeff draws balls from the jar below. He draws two balls without replacement. Find the probability that he draws a red ball and then a blue ball, in that order.

4 red

3 blue

2 yellow

Example: Selecting From an Jar of Balls

Jeff draws balls from the jar below. He draws two balls, this time with replacement. Find the probability that he gets a red and then a blue ball, in that order.

4 red

3 blue

2 yellow

Example:Applied Example: Color Blindness

In a test conducted by the U.S. Army, it was found that of 1000 new recruits (600 men and 400 women), 50 of the men and 4 of the women were red-green color-blind.Given that a recruit selected at random from this group is red-green color-blind, what is the probability that the recruit is a male?

Product Rule

Example: Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards.What is the probability that the first card drawn is an ace and the second card drawn is a face card?

Independent Events

  • If A and B are independent events, then

Test for the Independence of Two Events

  • Two events A and B are independent if and only if

Example: Consider the experiment consisting of tossing a fair coin twice and observing the outcomes.

Show that obtaining heads on the first toss and tails on the second toss are independent events.

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