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
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.