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

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

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

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

Example 9-1 (continued) d.the event C consisting of outcomes having a month that begins with M or N C = {March, May, November} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

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

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

Example 9-2 (continued) 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 9-2 (continued) 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definitions Impossible event: an event with no outcomes; has probability 0. Certain event: an event with probability 1. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

Example 9-3 (continued) 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

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

Complementary Events If A is an event and A is its complement, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

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

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 = 11 tees, and 2 tees are red, Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 9-4 (continued) 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)? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.