Probability The Basics – Section 4.2

Learning Targets 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.

Definitions Event: Any collection of results or outcomes of a procedure. Ex. Getting a sum of 4 when rolling 2 dice. {(1, 3), (3, 1), (2, 2)} Simple Event: An outcome or an event that cannot be further broken down into simpler components. Ex. An outcome result when rolling a die {3}, {6} Sample Space: Consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further. Ex. The outcomes of rolling a die {1, 2, 3, 4, 5, 6}

Definitions in-play Procedure Example of Event Complete Sample Space Single Birth 1 Female {f, m} 3 Births 2 Females and 1 Male {fff, ffm, fmf, mff, fmm, mfm, mmf, mmm} Example 1: In the following display, we use “f” to denote a female baby and “m” to denote a male baby. 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.

Notation for Probability P denotes a probability A, B, C denote specific events. P(A) denotes the probability of event A occurring.

Finding Probability – Method 1 Relative Frequency Approximation of Probability: Conduct or observe a procedure, and count the number of times that event A actually occurs. Based on these actual result, P(A) is approximated as follows:

Finding Probability – Method 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, Is the probability of passing a test ½ or 0.5 because you either pass the test or do not?

Denote A = a car in a crash this year 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. Denote A = a car in a crash this year

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:

Finding Probability – Method 3 Subjective Probabilities: P(A), the probability of event A, is estimated by using knowledge of relevant circumstances. Example: When trying to estimate the probability of an astronaut surviving a mission in a space shuttle, experts consider past event along with changes in technology to develop an estimate of probability. That probability has been estimated by NASA Scientists to 0.99.

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.

Caution – Important Notes 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. Round off probability results to 3 significant digits.

Possible Values for Probabilities

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

Practice A typical question on the SAT requires the test taker to select one of five possible choices: A, B, C, D, or E. Because only one answer is correct, if you make a random guess, your probability of being correct is 1/5. Find the probability of making a random guess and being incorrect. A – being incorrect P A = 4 5

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? 68 of the trees are large, let A - large: b) What is the probability that one tree selected at random is diseased? 37 of trees are diseased, let B - diseased:

Example 5: 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 795 + 10 = 805 total persons executed. Let W = a randomly selected execution was that of a woman. P(W) = 10/805 = 0.0124. Yes. Since 0.0124  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.

Example 6: 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

Example 6 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

Practice Pg. 148-150 #3, 5-7, 15, 18, 33

Practice Review

Round 1 When studying the affect of heredity on height, we can express each individual genotype, AA, Aa, aA, aa, on an index card and shuffle the four cards and randomly select one of them. What is the probability that we select a genotype in which the two components are different? 4 5

Round 2 The internet service provided AOL asked users this question about Kentucky Fried Chicken (KFC): “Will KFC gain or lose business after eliminating trans fats?” Among the responses received, 1941 said that KFC would gain business, 1260 said that KFC business would remain the same, and 204 said that KFC would lose business. Find the probability that a randomly selected response states that KFC would gain business. A – KFC gain business 𝑃 𝐴 = 1941 1941+1260+204 = 1941 3405

Round 3 If a year is selected at random, find the probability that Thanksgiving Day will be (a) on a Wednesday or (b) on a Thursday. (a) (b) 𝟎 𝟕 =𝟎 𝟏 𝟕

Round 4 Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child. Sample Space: {bbb, bbg, bgb, gbb, bgg, gbg, ggb, ggg} Favorite Outcomes: {bbg, bgb, gbb} P(exactly 2 boys) = 𝟑 𝟖

Round 5 The table below summarizes the test results for 98 different subjects. In each case, it was known whether or not the subject lied. Assume that one of the 98 test results is randomly selected, find the probability that it is a positive result. P(positive result) = Did the Subject Actually Lie? No (Did Not Lie) (Yes) Lied Positive Test Result (Polygraph test indicated that the subject lied) 15 (false positive) 42 (true positive) Negative Test Result (Polygraph test indicated that the subject did not lie). 32 (true negative) 9 (false negative) 𝟒𝟕 𝟗𝟖

Round 6 When predicting the chance that we will elect a Republican President in the year 2016, we could reason that there are two possible outcomes (Republican, not Republican), so the probability of a Republican president is ½ or 0.5. Is this reasoning correct? Why or why not? No, because there might have some other presidential candidates come from the other parties.

Round 7 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 84 + 16 = 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.

Homework Quiz Derrick Rose hit 197 of his 250 free throws in the 2012 season. What is his approximate probability of hitting a free throw?

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