Chapter Probability 9 9 Copyright © 2013, 2010, and 2007, Pearson Education, Inc

Chapter

Probability

99

Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

9-1 How Probabilities are Determined

Determining Probabilities Mutually Exclusive Events Complementary Events Non-Mutually Exclusive Events


Definitions

Experiment: an activity whose results can be observed and recorded.

Outcome: each of the possible results of an experiment.

Sample space: a set of all possible outcomes for an experiment.

Event: any subset of a sample space.


Example 9-1

Suppose an experiment consists of drawing 1 slip of paper from a jar containing 12 slips of paper, each with a different month of the year written on it. Find each of the following:

a. the sample space S for the experiment

S = {January, February, March, April, May, June, July, August, September, October, November, December}


Example 9-1 (continued)

b. the event A consisting of outcomes having a month beginning with J

A = {January, June, July}

c. the event B consisting of outcomes having the name of a month that has exactly four letters

B = {June, July}



d. the event C consisting of outcomes having a month that begins with M or N

C = {March, May, November}


Determining Probabilities

Experimental (empirical) probability: determined by observing outcomes of experiments.

Theoretical probability: the outcome under ideal conditions.

Equally likely: when one outcome is as likely as another

Uniform sample space: each possible outcome of the sample space is equally likely.


Law of Large Numbers (Bernoulli’s Theorem)

If an experiment is repeated a large number of times, the experimental (empirical) probability of a particular outcome approaches a fixed number as the number of repetitions increases.


Probability of an Event with Equally Likely Outcomes

For an experiment with sample space S with equally likely outcomes, the probability of an event A is


Example 9-2

Let S = {1, 2, 3, 4, 5, …, 25}. If a number is chosen at random, that is, with the same chance of being drawn as all other numbers in the set, calculate each of the following probabilities:

a. the event A that an even number is drawn

A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}, so n(A) = 12.



b. the event B that a number less than 10 and greater than 20 is drawn

c. the event C that a number less than 26 is drawn

C = S, so n(C) = 25.



d. the event D that a prime number is drawn

e. the event E that a number both even and prime is drawn

D = {2, 3, 5, 7, 11, 13, 17, 19, 23}, so n(D) = 9.

E = {2}, so n(E) = 1.


Definitions

Impossible event: an event with no outcomes; has probability 0.

Certain event: an event with probability 1.


Probability Theorems

If A is any event and S is the sample space, then

The probability of an event is equal to the sum of the probabilities of the disjoint outcomes making up the event.


Example 9-3

If we draw a card at random from an ordinary deck of playing cards, what is the probability that

a. the card is an ace?

There are 52 cards in a deck, of which 4 are aces.



If we draw a card at random from an ordinary deck of playing cards, what is the probability that

b. the card is an ace or a queen?

There are 52 cards in a deck, of which 4 are aces and 4 are queens.


Mutually Exclusive Events

Events A and B are mutually exclusive if they have no elements in common; that is,

For example, consider one spin of the wheel.S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 1, 2, 3, 4}, and B = {5, 7}.

If event A occurs, then event B cannot occur.


Mutually Exclusive Events

If events A and B are mutually exclusive, then

The probability of the union of events such that any two are mutually exclusive is the sum of the probabilities of those events.


Complementary Events

Two mutually exclusive events whose union is the sample space are complementary events.

For example, consider the event A = {2, 4} of tossing a 2 or a 4 using a standard die. The complement of A is the set A = {1, 3, 5, 6}.

Because the sample space is S = {1, 2, 3, 4, 5, 6},


Complementary Events

If A is an event and A is its complement, then


Non-Mutually Exclusive Events

Let E be the event of spinning an even number.

E = {2, 14, 18}

Let T be the event of spinning a multiple of 7.

T = {7, 14, 21}


Summary of Probability Properties

1. P(Ø) = 0 (impossible event)

2. P(S) = 1, where S is the sample space (certain event).

3. For any event A, 0 ≤ P(A) ≤ 1.


Summary of Probability Properties

4. If A and B are events and A ∩ B = Ø, then P(A U B) = P(A) + P(B).

5. If A and B are any events, then P(A U B) = P(A) + P(B) − P(A ∩ B).

6. If A is an event, then P(A) = 1 − P(A).


Example 9-4

A golf bag contains 2 red tees, 4 blue tees, and 5 white tees.

a. What is the probability of the event R that a tee drawn at random is red?

Because the bag contains a total of 2 + 4 + 5 = 11

tees, and 2 tees are red,



b. What is the probability of the event “not R”; that is a tee drawn at random is not red?

c. What is the probability of the event that a tee drawn at random is either red (R) or blue (B); that is, P(R U B)?


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Chapter Probability 9 9 Copyright © 2013, 2010, and 2007, Pearson Education, Inc