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Section 4-1 Review and Preview

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**Preview Rare Event Rule for Inferential Statistics:**

If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Statisticians use the rare event rule for inferential statistics.

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**Basic Concepts of Probability**

Section 4-2 Basic Concepts of Probability

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Key Concept This section presents three approaches to finding the probability of an event. The most important objective of this section is to learn how to interpret probability values.

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Part 1 Basics of Probability

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**Events and Sample Space**

Event: any collection of results or outcomes of a procedure. Simple Event: an outcome or an event that cannot be further broken down into simpler components. Sample Space: for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further.

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**Example 1: In the following display, we use “f” to denote a female baby and “m”**

to denote a male baby. Procedure Example of Event Complete Sample Space Single birth 1 female (simple event) {f, m} 3 births 2 females and 1 male {fff, ffm, fmf, fmm, mff, (ffm, fmf, mff, are all mfm, mmf, mmm} simple events resulting in 2 females and a male) With one birth, the result of 1 female is a simple event because it cannot be broken down any further. With three births, the event “2 females and 1 male” is not a simple event because it can be broken down into simpler events, such as ffm, fmf, or mff. With three births, the sample space consists of the 8 simple events listed Above. With three births, the outcome if ffm is considered a simple event, it is an outcome that cannot be broken down any further. We might incorrectly that ffm can be broken down into the individual results of f, f, and m, but f, f, and m are not individual outcomes from three births. With three births there are exactly 8 outcomes that are simple events: fff, ffm, fmf, fmm, mff, mfm, mmf, and mmm.

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**Notation for Probabilities**

P - denotes a probability. A, B, and C - denote specific events. P(A) - denotes the probability of event A occurring.

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**Computing Probability**

Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is approximated as follows:

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Example 2: For a recent year there were 6,511,100 cars that crashed among the 135,670,000 cars registered in the United States (based on data from Statistical Abstract of the United States). Find the probability that a randomly selected car in the United States will be in a crash this year.

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**Computing Probability - continued**

Basic Rules for Computing Probability - continued Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then

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Example 3: There are 8 equally likely outcomes when a woman has three babies. What is the probability that she has 3 girls? The possible outcomes are: GGG GGB GBB GBG BBB BBG BGG BGB Only one of these possibilities have all three girls, so:

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**Computing Probability - continued**

Basic Rules for Computing Probability - continued Rule 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

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Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

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Probability Limits Always express a probability as a fraction or decimal number between 0 and 1. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. For any event A, the probability of A is between 0 and 1 inclusive. That is, 0 P(A) 1.

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**Possible Values for Probabilities**

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Complementary Events The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur.

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**Rounding Off Probabilities**

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)

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**Example 4: Some trees in a forest were showing signs of disease**

Example 4: Some trees in a forest were showing signs of disease. A random sample of 200 trees of various sizes was examined yielding the following results: a) What is the probability that one tree selected at random is large? There are a total of 200 trees and 68 of them are large, so: b) What is the probability that one tree selected at random is diseased? There are a total of 200 trees and 37 of them are diseased, so:

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Example 4 continued: Some trees in a forest were showing signs of disease. A random sample of 200 trees of various sizes was examined yielding the following results: c) What is the probability that one tree selected at random is both small and diseased? There are a total of 200 trees and 8 of them are small AND diseased, so: d) What is the probability that one tree selected at random is both doubtful and medium? There are a total of 200 trees and 32 of them are medium AND doubtful, so:

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Example 5: Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive. In one of New York State’s instant lottery games, the chance of a win are stated as “4 in 21”.

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Example 6: Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive. If you make a random guess for the answer to a true/false test question, there is a chance of being correct.

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Example 7: Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive. It is impossible to get five aces when selecting cards from a shuffled deck.

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**Example 8: A couple decides to have three children**

Example 8: A couple decides to have three children. Find the probability of each event. a) There is exactly one boy. Let’s recall our sample space: GGG GGB GBB GBG BBB BBG BGG BGB Of those 8 possibilities, 3 of them have exactly one boy, so: b) There are exactly two boys. Of the 8 possibilities from above, 3 of them have exactly two boys, so: c) All are boys. Of the 8 possibilities from above, 1 of them have all boys, so:

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Example 9: The 110th Congress of the United States included 84 male Senators and 16 female Senators. If one of these Senators is selected, what is the probability that a woman is selected? Does this probability agree with a claim that men and women have the same chance of being elected as Senators? There are = 100 total senators. Let W = selecting a woman. P(W) = 16/100 = 0.16. No, this probability is too far below 0.50 to agree with the claim that men and women have equal opportunities to become a senator.

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Example 10: In the last 30 years, death sentence executions in the United States included 795 men and 10 women (based on data from the Associated Press). If an execution is randomly selected, find the probability that the person executed is a woman. Is it unusual for a woman to be executed? How might this discrepancy be explained? There were = 805 total persons executed. Let W = a randomly selected execution was that of a woman. P(W) = 10/805 = Yes. Since 0.05, it is unusual for an executed person to be a woman. This is due to the fact that more crimes worthy of the death penalty are committed by men than women. “Rare event rule for inferential statistics” from Section 4.1.

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Example 11: Each of two parents has the genotype brown/blue, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one brown allele, that color will dominate and the eyes will be brown. (The actual determination of eye color is somewhat more complicated.) a) List the different possible outcomes. Assume that those outcomes are equally likely. Listed with the father’s contribution first, the sample space has 4 simple events: brown/brown brown/blue blue/brown blue/blue b) What is the probability that a child of these parents will have blue/blue genotype? P(blue/blue) = ¼ = 0.25

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Example 11 continued: c) What is the probability that a child will have brown eyes? P(brown eyes) = P(brown/brown or brown/blue or blue/brown) = ¾ = 0.75

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**Beyond the Basics of Probability: Odds**

Part 2 Beyond the Basics of Probability: Odds

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Odds The actual odds against event A occurring are the ratio P(A)/P(A), usually expressed in the form of a:b (or “a to b”), where a and b are integers having no common factors. The actual odds in favor of event A occurring are the ratio P(A)/P(A), which is the reciprocal of the actual odds against the event. If the odds against A are a:b, then the odds in favor of A are b:a. The payoff odds against event A occurring are the ratio of the net profit (if you win) to the amount bet. payoff odds against event A = (net profit) : (amount bet)

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Example 12: When the horse Barbaro won the 132nd Kentucky Derby, a $2 bet that Barbaro would win resulted in a return of $14.20. a) How much net profit was made from a $2 win bet in Barbaro? If a $2 bet results in a net return of $14.20, the net profit is – 2.00 = $12.20. b) What were the payoff odds against a Barbaro win? If you get back your bet plus $12.20 for every $2 you bet, the payoff odds are 12.2 : 2 (We express this as 6.1 : 1 or 61 : 10), or approximately 6 : 1.

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Example 12 continued: When the horse Barbaro won the 132nd Kentucky Derby, a $2 bet that Barbaro would win resulted in a return of $14.20. c) Based on preliminary wagering before the race, bettors collectively believed that Barbaro had a 57/500 probability of winning. Assuming that 57/500 was a true probability of s Barbaro victory, what were the actual odds against his winning? Odds against W = P(not win)/P(W) = (443/500)/(57/500) = 443/57 or 443 : 57. About 8 : 1. d) If the payoff odds were the actual odds found in part (c), how much would a $2 win ticket be worth after the Barbaro win? If the payoff odds are 7.77 : 1, a win gets back your bet plus $7.77 for every $1 bet. If you bet $2 and win, you get back 2 + $ The ticket would be worth $17.54 and your profit would be – 2.00 = $15.54.

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