06-Random Variables complete

00000095 noneoftheshopperswillmakeapurchase

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Unformatted text preview: correct. n is fixed in advance at 10, indep is ok (picking answer at random), 2 outcomes (correct, not correct), BUT p = 0.50 or p = 0.25. c. Four students are randomly picked without replacement from large student body listing of 1000 women and 1000 men. X = number of women among the four selected students. n is fixed in advance at 4, indep is ok (since a large population), 2 outcomes (W, M), and p = 0.50. What if the student body listing consisted only of 10 women and 10 men? p will not be 0.50 on each trial if N is small, and independence not ok. Rule of Thumb: population at least 10 times as large as the sample ok! 47 The Binomial Formula We will develop the formula together using our probability knowledge. Suppose that of the online shoppers for a particular website that start filling a shopping cart with items, 25% actually make a purchase (complete a transaction). We have a random sample of 10 such online shoppers. If the stated rate is true, what is the probability that ... ... all 10 shoppers will actually make a purchase? P(1st P and 2nd P and … and 10th P) = (0.25)10 = 0.00000095 ... none of the shoppers will make a purchase? P(1st NP and 2nd NP and … and 10th NP) = (0.75)10 = 0.0563 ... just 1 shopper will make a purchase? X __ __ __ __ __ __ __ __ __ __ => 0.25(0.75)9 = 0.01877 __ X __ __ __ __ __ __ __ __ __ => 0.25(0.75)9 = 0.01877 … 10 of these so 0.1877 __ __ __ __ __ __ __ __ __ __ X => 0.25(0.75)9 = 0.01877 Do you know that with only the basic probability knowledge from chapter 7, you just calculated three binomial probabilities that are based on the following formula? The binomial distribution: Probability of exactly k successes in n trials … n P ( X k ) p k (1 p ) n k k n n! where k k! ( n k )! for k 0,1,2,..., n (this represents the number of ways to select k items from n) 48 Try it! The first part … n You can think of the computation of in the following way … Suppose you had n friends, k how many ways could you invite k to dinner? The ones “at the ends” are easy to do without even using the formula or a calculator. Your calculator is likely to have this complete function or at least a factorial ! option. On many calculators this combinations function is found under the math probability menu and expressed as nCr. 1. 10 1 (invite no one) 0 2. 10 1 (invite all) 10 10 10 1 4. 10 10 (leave 1 at home) 9 3. 5. 10 10! 10(9)(8) (2)(1) 90 2 2!8! (2)(1)(8)(7) (2)(1) 2 45 Try it! Finding Binomial Probabilities Recall we have a random sample of n = 10 online shoppers from a large population of such shoppers and that p = 0.25 is the population proportion who actually make a purchase. a. What is the probability of selecting exactly one shopper who actually makes a purchase? 10 P(X = 1) = (0.25)1(0.75)9 = 10(0.25)(0.075085) = 0.1877 1 b. What is the probability of selecting exactly two shoppers who actually make a purchase? 10 P(X = 2) = (0.25)2(0.75)8 = 45(0.0625)(0.10011) = 0.2816 2 c. What is the probability of selecting at least one shopper...
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