© 2010 Pearson Education, Inc. All rights reserved Chapter 9 9 Probability

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Data Analysis and Probability  K–2: Children should discuss events related to their experience as likely or unlikely. (p. 400)  3–5: Children should be able to “describe events as likely or unlikely and discuss the degree of likelihood using words such as certain, equally likely, and impossible.” They should be able to “predict the probability of outcomes of simple experiments and test the predictions.” They should “understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.” (p. 400)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Data Analysis and Probability 6–8: Children should “understand and use appropriate terminology to describe complementary and mutually exclusive events.” They should be able “to make and test conjectures about the results of experiments and simulations.” They should be able to “compute probabilities of compound events using methods such as organized lists, tree diagrams, and area models.” (p. 401)

Slide Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 9-1 How Probabilities are Determined  Determining Probabilities  Mutually Exclusive Events  Complementary Events  Non-Mutually Exclusive Events

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 or repetitions increases.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Definitions Impossible event: an event with no outcomes; has probability 0. Certain event: an event with probability 1.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Complementary Events If A is an event and A is its complement, then

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-5 Find the probability of rolling a sum of 7 or 11 when rolling a pair of fair dice. There are 36 possible rolls.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-5 (continued) There are 6 ways to form a sum of “7”, so There are 6 ways to form a sum of “11”, so

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-5 (continued) The sample space for the experiment is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, but the sample space is not uniform; i.e., the probabilities of the given sums are not equal. The probability of rolling a sum of 7 or 11 is

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-6 A fair pair of dice is rolled. Let E be the event of rolling a sum that is an even number and F the event of rolling a sum that is a prime number. Find the probability of rolling a sum that is even or prime, that is, P(E U F). E U F = {2, 4, 6, 8, 10, 12, 3, 5, 7, 11}

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-6 (continued)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-6 (continued) Alternate solution 1: E = {2, 4, 6, 8, 10, 12} and F = {2, 3, 5, 7, 11}. Thus, E and F are not mutually exclusive because E ∩ F = {2}.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-6 (continued) Alternate solution 2: E = {2, 4, 6, 8, 10, 12} and F = {2, 3, 5, 7, 11}. Thus, E U F = {2, 3, 4, 5, 6, 7, 8, 10, 11} and E U F = {9}.