Slide Copyright © 2009 Pearson Education, Inc. Chapter 7 Probability

Slide Copyright © 2009 Pearson Education, Inc. Definitions An experiment is a controlled operation that yields a set of results.  We don’t know the outcome in advance! The possible results of an experiment are called its outcomes. An event is a subcollection of the outcomes of an experiment. Example – experiment: rolling a die.  Outcomes: 1, 2, 3, 4, 5, 6  Some events: rolling an even number, rolling a 1 or a 6.

Slide Copyright © 2009 Pearson Education, Inc. Definitions continued Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment.

Slide Copyright © 2009 Pearson Education, Inc. Empirical Probability Example: In 100 tosses of a fair die, 19 landed showing a 3. Find the empirical probability of the die landing showing a 3. Let E be the event of the die landing showing a 3.

Slide Copyright © 2009 Pearson Education, Inc. The Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

Slide Copyright © 2009 Pearson Education, Inc. 7.2 Theoretical Probability Equally likely outcomes If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes. For equally likely outcomes, the probability of Event E may be calculated with the following formula.

Slide Copyright © 2009 Pearson Education, Inc. Example A die is rolled. Find the probability of rolling a) a 2. b) an odd number. c) a number less than 4. d) an 8. e) a number less than 9.

Slide Copyright © 2009 Pearson Education, Inc. A die is rolled. Find the probability of rolling: a) a 2. b) an odd number. There are three ways an odd number can occur 1, 3 or 5. c) a number less than 4. Three numbers are less than 4.

Slide Copyright © 2009 Pearson Education, Inc. d) an 8. There are no outcomes that will result in an 8. e) a number less than 9. All outcomes are less than 9. The event must occur and the probability is 1. A die is rolled. Find the probability of rolling:

Slide Copyright © 2009 Pearson Education, Inc.

Slide Copyright © 2009 Pearson Education, Inc. Important Facts The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1 inclusive; that is, 0 <= P(E) <= 1. The sum of the probabilities of all possible outcomes of an experiment is 1.

Slide Copyright © 2009 Pearson Education, Inc. Example A standard deck of cards is well shuffled. Find the probability that the card is selected. a) a 10. b) not a 10. c) a card greater than 4 and less than 7.

Slide Copyright © 2009 Pearson Education, Inc. Example continued a) a 10 There are four 10’s in a deck of 52 cards. b) not a 10

Slide Copyright © 2009 Pearson Education, Inc. Example continued c) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s, and 6’s.

Slide Copyright © 2009 Pearson Education, Inc. 7.4 Expected Value The symbol P 1 represents the probability that the first event will occur, and A 1 represents the net amount won or lost if the first event occurs.

Slide Copyright © 2009 Pearson Education, Inc. Example Calvin’s expectation is -$2.50 when he purchases one ticket. When Calvin Winters attends a tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket

Slide Copyright © 2009 Pearson Education, Inc. Fair Price Example: Suppose you are playing a game in which you spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If is costs $10 to play the game, determine a) the expectation of the person who plays the game. b) the fair price to play the game. $10 $2 $20$15 Fair price = expected value + cost to play

Slide Copyright © 2009 Pearson Education, Inc. Solution $0 3/8 $10 $5-$8Amount Won/Lost 1/8 3/8Probability $20$15$2Amt. Shown on Wheel $10 $2 $20$15

Slide Copyright © 2009 Pearson Education, Inc. Solution Fair price = expectation + cost to play = -$ $10 = $8.87 Thus, the fair price is about $8.87.

Slide Copyright © 2009 Pearson Education, Inc. Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M N distinct ways. Tree diagrams are helpful for visualizing the different outcomes.

Slide Copyright © 2009 Pearson Education, Inc. Example Two balls are to be selected without replacement from a bag that contains one purple, one blue, and one green ball. a)Use the counting principle to determine the number of possible outcomes. b)Construct a tree diagram and list the outcomes. a) 3 2 = 6 ways b) B P B G B G P G P PB PG BP BG GP GB

Slide Copyright © 2009 Pearson Education, Inc. Example Two balls are to be selected without replacement from a bag that contains one purple, one blue, and one green ball. c)Find the probability that one blue ball is selected. d)Find the probability that a purple ball followed by a green ball is selected. c) d) B P B G B G P G P PB PG BP BG GP GB

Slide Copyright © 2009 Pearson Education, Inc. Example Two balls are to be selected without replacement from a bag that contains one purple, one blue, and one green ball. e)Find the probability that a blue ball is NOT selected. e) B P B G B G P G P PB PG BP BG GP GB

Slide Copyright © 2009 Pearson Education, Inc. And Problems P(A and B) = P(A) P(B) Example: Two cards are to be selected with replacement from a deck of cards. Find the probability that two red cards will be selected.

Slide Copyright © 2009 Pearson Education, Inc. Example Two cards are to be selected without replacement from a deck of cards. Find the probability that two red cards will be selected.

Slide Copyright © 2009 Pearson Education, Inc. Or Problems P(A or B) = P(A) + P(B) - P(A and B) One card is selected from a standard deck of playing cards. Determine the probability of the following events. a) selecting a 3 or a jack b) selecting a jack or a heart c) selecting a picture card or a red card d) selecting a red card or a black card

Slide Copyright © 2009 Pearson Education, Inc. Solutions a) 3 or a jack b) jack or a heart

Slide Copyright © 2009 Pearson Education, Inc. Solutions continued c) picture card or red card d) red card or black card

Slide Copyright © 2009 Pearson Education, Inc. Odds in Favor

Slide Copyright © 2009 Pearson Education, Inc. Example Find the odds in favor of landing on blue in one spin of the spinner. The odds in favor of spinning blue are 3:5.

Slide Copyright © 2009 Pearson Education, Inc. 7.3 Odds Odds Against

Slide Copyright © 2009 Pearson Education, Inc. Independent Events Event A and Event B are independent events if the occurrence of either event in no way affects the probability of the occurrence of the other event. Experiments done with replacement will result in independent events, and those done without replacement will result in dependent events.

Slide Copyright © 2009 Pearson Education, Inc. Example A package of 30 tulip bulbs contains 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. Three bulbs are randomly selected and planted. Find the probability of each of the following. a.All three bulbs will produce pink flowers. b.The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower. c.None of the bulbs will produce a yellow flower. d.At least one will produce yellow flowers.

Slide Copyright © 2009 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. a. All three bulbs will produce pink flowers.

Slide Copyright © 2009 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. b. The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower.

Slide Copyright © 2009 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. c. None of the bulbs will produce a yellow flower.

Slide Copyright © 2009 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. d. At least one will produce yellow flowers. P(at least one yellow) = 1 - P(no yellow)