© 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. 9-2Multistage Experiments with Tree Diagrams and Geometric Probabilities  More Multistage Experiments  Independent Events  Modeling Games  Geometric Probability (Area Models) Revisited

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tree Diagrams Suppose the spinner is spun twice. The tree diagram shows the possible outcomes.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tree Diagrams The sample space can be written {BB, BR, BY, RB, RR, RY, YB, YR, YY}. The probability of each outcome is

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tree Diagrams Alternatively, we can generate the sample space using a table. YB

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-7 Suppose we toss a fair coin 3 times and record the results. Find each of the following: a.The sample space for this experiment. S = {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT} b.The event A of tossing 2 heads and 1 tail A = {HHT, HTH, THH}

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-7 (continued) c.The event B of tossing no tails. B = {HHH} d.The event C of tossing a head on the last toss C = {HHH, HTH, THH, TTH}

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide More Multistage Experiments The box in figure (a) contains one colored ball and two white balls. If a ball is drawn at random and the color recorded, a tree diagram for the experiment might look like the one in figure (b). Because each ball has the same chance of being drawn, we may combine the branches and obtain the tree diagram shown in figure (c).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Area Model If the area of the entire rectangle is 1, the area of the white portion depicting the probability of drawing a white ball is represented as of the entire rectangle. The cross-hatched portion of the figure represents the probability of choosing a colored ball, or

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Suppose a ball is drawn at random from the box in the figure and its color recorded. The ball is then replaced, and a second ball is drawn and its color recorded. The sample space for this two-stage experiment may be recorded using ordered pairs as

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide The tree diagram for this experiment is shown below.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Multiplication Rule for Probabilities for Tree Diagrams For all multistage experiments, the probability of the outcome along any path of a tree diagram is equal to the product of all the probabilities along the path.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Suppose two balls are drawn one-by-one without replacement.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Independent Events Independent events: when the outcome of one event has no influence on the outcome of a second event.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Independent Events If two coins are flipped and event E 1 is obtaining a head on the first coin and E 2 is obtaining a tail on the second coin, then E 1 and E 2 are independent events because one event has no influence on possible outcomes.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Independent Events

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Independent Events For any independent events E 1 and E 2,

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-8 The figure shows a box with 11 letters. Some letters are repeated. Suppose 4 letters are drawn at random from the box one-by-one without replacement. What is the probability of the outcome BABY, with the letters chosen in exactly the order given?

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-8 (continued) We are interested in only the tree branch leading to the outcome BABY. The probability of the first B is because there are 2 B’s out of 11 letters. The probability of the second B is because there are 9 letters left after 1 B and 1 A have been chosen.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 9-9 A letter is drawn from box 1 and placed in box 2. Then, a letter is drawn from box 2 and placed in box 3. Finally, a letter is drawn from box 3. What is the probability that the letter drawn from box 3 is B? (Call this event E.)

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide There are two colored marbles and one white marble in a box. Gwen mixes the marbles, and Arthur draws two marbles at random without replacement. If the two marbles match, Arthur wins; otherwise, Gwen wins. Does each player have an equal chance of winning? Modeling Games

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Modeling Games The probability that the marbles are the same color is while the probability that they are not the same color is Thus, the players do not have the same chance of winning.