Probability Review. The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE

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1 Probability Review The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE

2 Probability Models In Section 5.1, we used simulation to imitate chance behavior. Fortunately, we don t have to always rely on simulations to determine the probability of a particular outcome. Descriptions of chance behavior contain two parts: Definition: The sample space S of a chance process is the set of all possible outcomes. Probability Rules A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

3 Example: Roll the Dice Give a probability model for the chance process of rolling two fair, six-sided dice one that s red and one that s green. Probability Rules Sample Space 36 Outcomes Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36.

4 Probability Models Probability models allow us to find the probability of any collection of outcomes. Definition: An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. Probability Rules If A is any event, we write its probability as P(A). In the dice-rolling example, suppose we define event A as sum is 5. There are 4 outcomes that result in a sum of 5. Since each outcome has probability 1/36, P(A) = 4/36. Suppose event B is defined as sum is not 5. What is P(B)? P(B) = 1 4/36 = 32/36

5 Basic Rules of Probability All probability models must obey the following rules: The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities whose sum is 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula Probability Rules P(A) number of outcomes corresponding to event A total number of outcomes in sample space The probability that an event does not occur is 1 minus the probability that the event does occur. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Definition: Two events are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together.

6 Basic Rules of Probability For any event A, 0 P(A) 1. If S is the sample space in a probability model, P(S) = 1. Probability Rules In the case of equally likely outcomes, P(A) number of outcomes corresponding to event A total number of outcomes in sample space Complement rule: P(A C ) = 1 P(A) P( at least 1 ) = 1 P( none )

7 Basic Rules of Probability General Addition Rule: P(A or B) = P(A) + P(B) P(A and B) Remember that if A & B are mutually exclusive, then P(A and B) = 0 Probability Rules General Multiplication Rule: P(A B) = P(A) P(B A) Remember that if A & B are independent, then P(B A) = P(B) Find these two Rules on the Formula Sheet!

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9 Solution: Using the general mult. rule: 0.18 = 0.6 * P(E D), so P(E D) = 0.3 And, since D & E are independent, P(E) = P(E D) = 0.3 Then, using the general addition rule: P (D U E) = = 0.72 The correct answer is D.

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11 Solution: (.30)(.15) + (.50)(.25) = =.170 The correct answer is C

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13 Since there is no association between age and preferred method of eating ice cream, the two categories are Independent. Thus, P(Adult Cone) = P(Adult Cup) = P(Adult) = 1/3 & P(Child Cone) = P(Child Cup) = P(Child) = 2/3 The correct answer is C

14 Two-Way Tables and Probability Note, the previous example illustrates the fact that we can t use the addition rule for mutually exclusive events unless the events have no outcomes in common. The Venn diagram below illustrates why. Probability Rules General Addition Rule for Two Events If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) P(A and B)

15 Venn Diagrams and Probability Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed. The complement A C contains exactly the outcomes that are not in A. Probability Rules The events A and B are mutually exclusive (disjoint) because they do not overlap. That is, they have no outcomes in common.

16 Venn Diagrams and Probability The intersection of events A and B (A B) is the set of all outcomes in both events A and B. Probability Rules The union of events A and B (A B) is the set of all outcomes in either event A or B. Hint: To keep the symbols straight, remember for union and for intersection.

17 General Multiplication Rule The idea of multiplying along the branches in a tree diagram leads to a general method for finding the probability P(A B) that two events happen together. General Multiplication Rule The probability that events A and B both occur can be found using the general multiplication rule P(A B) = P(A) P(B A) where P(B A) is the conditional probability that event B occurs given that event A has already occurred. Conditional Probability and Independence

18 Example: Who Visits YouTube? See the example on page 320 regarding adult Internet users. What percent of all adult Internet users visit video-sharing sites? P(video yes 18 to 29) = = P(video yes 30 to 49) = = P(video yes 50 +) = = P(video yes) = =

19 Independence: A Special Multiplication Rule When events A and B are independent, we can simplify the general multiplication rule since P(B A) = P(B). Definition: Multiplication rule for independent events If A and B are independent events, then the probability that A and B both occur is P(A B) = P(A) P(B) Example: Following the Space Shuttle Challenger disaster, it was determined that the failure of O-ring joints in the shuttle s booster rockets was to blame. Under cold conditions, it was estimated that the probability that an individual O-ring joint would function properly was Assuming O-ring joints succeed or fail independently, what is the probability all six would function properly? P(joint1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5 OK and joint 6 OK) =P(joint 1 OK) P(joint 2 OK) P(joint 6 OK) =(0.977)(0.977)(0.977)(0.977)(0.977)(0.977) = 0.87 Conditional Probability and Independence

20 Section 6.1 Discrete vs. Continuous Random Variables Discrete Random Variables X is defined as a fixed set of possible values with gaps in between Examples: grades on a 4 point scale X={0, 1, 2, 3, 4} Height (to the nearest cm) X= {151, 157, 165, 173, 185, 198} Sum of two dice X={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Normal Body Temperature in F X={97.7, 98.0, 98.6, 99.1, 99.5} Continuous Random Variables X is defined as all values in an interval of numbers Examples: X = GPAs of high school seniors (1.8 < X < 4.6) X = Height of 6 th per AP Stats students in cm (151 < X < 198) X = length of an FHS varsity football game (hours) (1.25 < X < 3) X = Body Temp in F (97.7 < X < 99.5)

21 Section 6.1 Discrete vs. Continuous Random Variables Continuous Random Variables Probability distribution is described by a density curve and the probability P(A) of any event A is the area under the curve Total Area under curve = 1 Event A must be an interval of X

22 Binomial Settings When the same chance process is repeated several times, we are often interested in whether a particular outcome does or doesn t happen on each repetition. In some cases, the number of repeated trials is fixed in advance and we are interested in the number of times a particular event (called a success ) occurs. If the trials in these cases are independent and each success has an equal chance of occurring, we have a binomial setting. Definition: A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a binomial setting are B I N S Binary? The possible outcomes of each trial can be classified as success or failure. Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. Number? The number of trials n of the chance process must be fixed in advance. Success? On each trial, the probability p of success must be the same. Binomial and Geometric Random Variables

23 Binomial Probability The binomial coefficient counts the number of different ways in which k successes can be arranged among n trials. The binomial probability P(X = k) is this count multiplied by the probability of any one specific arrangement of the k successes. Binomial Probability If X has the binomial distribution with n trials and probability p of success on each trial, the possible values of X are 0, 1, 2,, n. If k is any one of these values, Number of arrangements of k successes P(X k) n p k (1 p) n k k Probability of k successes Probability of n-k failures Binomial and Geometric Random Variables

24 Geometric Settings In a binomial setting, the number of trials n is fixed and the binomial random variable X counts the number of successes. In other situations, the goal is to repeat a chance behavior until a success occurs. These situations are called geometric settings. Definition: A geometric setting arises when we perform independent trials of the same chance process and record the number of trials until a particular outcome occurs. The four conditions for a geometric setting are B I T S Binary? The possible outcomes of each trial can be classified as success or failure. Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. Trials? The goal is to count the number of trials until the first success occurs. Success? On each trial, the probability p of success must be the same. Binomial and Geometric Random Variables

25 Example: The Birthday Game Read the activity on page 398. The random variable of interest in this game is Y = the number of guesses it takes to correctly identify the birth day of one of your teacher s friends. What is the probability the first student guesses correctly? The second? Third? What is the probability the k th student guesses corrrectly? Verify that Y is a geometric random variable. B: Success = correct guess, Failure = incorrect guess I: The result of one student s guess has no effect on the result of any other guess. T: We re counting the number of guesses up to and including the first correct guess. S: On each trial, the probability of a correct guess is 1/7. Calculate P(Y = 1), P(Y = 2), P(Y = 3), and P(Y = k) P(Y 1) 1/7 P(Y 2) (6/7)(1/7) P(Y 3) (6/7)(6/7)(1/7) Notice the pattern? Geometric Probability If Y has the geometric distribution with probability p of success on each trial, the possible values of Y are 1, 2, 3,. If k is any one of these values, P(Y k) (1 p) k 1 p