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## SEVENTH EDITION and EXPANDED SEVENTH EDITION

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**Chapter 12**Probability**12.1**The Nature of Probability**Definitions**• An experiment is a controlled operation that yields a set of results. • The possible results of an experiment are called its outcomes. • An event is a subcollection of the outcomes of an experiment.**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 outcome that can occur for the given experiment.**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. Empirical Probability**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.**12.2**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.**Example**• A die is rolled. Find the probability of rolling • a) a 3. • b) an odd number • c) a number less than 4 • d) a 8. • e) a number less than 9.**Solutions: There are six equally likely outcomes: 1, 2, 3,**4, 5, and 6. • a) • b) There are three ways an odd number can occur 1, 3 or 5. • c) Three numbers are less than 4.**Solutions: There are six equally likely outcomes: 1, 2, 3,**4, 5, and 6. continued • d) There are no outcomes that will result in an 8. • e) All outcomes are less than 10. The event must occur and the probability is 1.**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.**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 heart. • d) a ace, one or 2. • e) diamond and spade. • f) a card greater than 4 and less than 7.**a) a 10**There are four 10’s in a deck of 52 cards. b) not a 10 Example continued**c) a heart**There are 13 hearts in the deck. d) an ace, 1 or 2 There are 4 aces, 4 ones and 4 twos, or a total of 12 cards. Example continued**d) diamond and spade**The word and means both events must occur. This is not possible. e) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s, and 6’s. Example continued**12.3**Odds**Odds Against**• Example: Find the odds against rolling a 5 on one roll of a die. The odds against rolling a 5 are 5:1.**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.**Probability from Odds**• Example: The odds against spinning a blue on a certain spinner are 4:3. Find the probability that • a) a blue is spun. • b) a blue is not spun.**Since the odds are 4:3 the denominators must be 4 + 3 = 7.**The probabilities ratios are: Solution**12.4**Expected Value (Expectation)**Expected Value**• E = x1P(x1) + x2P(x2) + x3P(x3) +…+ xnP(xn) or E = P(x1)x1 + P(x2)x2 + P(x3)x3 +…+ P(xn)xn (order doesn’t matter) • The symbol P1 represents the probability that the first event will occur, and A1 represents the net amount won or lost if the first event occurs.**Example**• Teresa is taking a multiple-choice test in which there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank. • If Teresa does not know the correct answer to a question, is it to her advantage or disadvantage to guess? • If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?**Solution**• Expected value if Teresa guesses.**Example: Winning a Prize**• When Calvin Winters attends a tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize.**Example**• 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.**Solution**• Calvin’s expectation is $2.50 when he purchases one ticket.**Fair Price**Fair price = expected value + cost to play**$10**$2 $15 $20 $10 $2 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.**Amt. Shown on Wheel**$2 $10 $15 $20 Probability 3/8 3/8 1/8 1/8 Amt. Won/Lost $8 $0 $5 $10 Solution**Solution**• Fair price = expectation + cost to play = $1.13 + $10 = $8.87 Thus, the fair price is about $8.87.**12.5**Tree Diagrams**Counting Principle**• If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct way, then the two experiments in that specific order can be performed in M• N ways. (Note: Order is important.)**Definitions**• Sample space: A list of all possible outcomes of an experiment. • Sample point (simple event): Each individual outcome in the sample space.**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 points in the sample space. • b) Construct a tree diagram and list the sample space. • 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.**a) 3 • 2 = 6 ways**b) c) d) P(purple then green = 1/6 Note: The probability of each simple event is PB B P PG G BP P B BG G GP P G GB B Solutions**12.6**Or and And Problems**Or Problems**• P(A or B) = P(A) + P(B) P(A and B) • Example: Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains an even number or a number greater than 5.**Solution**• P(A or B) = P(A) + P(B) P(A and B) • Thus, the probability of selecting an even number or a number greater than 5 is 7/10.**Example**• Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains a number less than 3 or a number greater than 7.**Solution**• There are no numbers that are both less than 3 and greater than 7. Therefore,**Mutually Exclusive**• Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.**Example**• 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**Solutions**• a) 3 or a jack • b) jack or a heart**Solutions continued**• c) picture card or red card • d) red card or black card**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.