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Probability Chapter 11 1 Counting Techniques and Probability Section 11.2 2 Examples Helen and Patty both belong to a club of 25 members. A committee of 4 is to be selected at random from the 25 members. Find the probability that both Helen and Patty will be selected. 2nCr2 23nCr2 P ( both selected) 25nCr4 253 12,650 1 50 3 Examples A piggy bank contains 2 quarters, 3 nickels, and 2 dimes. A person takes 2 coins at random from this bank. Label the coins Q1, Q2, N1, N2, N3, D1, and D2 so they can all be regarded as different. Then find the probabilities that the values of the 2 coins selected are the following: a. 35¢ b. 50¢ 2nCr1 2nCr1 2 2 4 a. P (35¢ ) 7 nCr 2 21 21 2nCr 2 1 b. P (50¢ ) 7 nCr 2 21 4 Examples Assume that 2 cards are drawn in succession and without replacement from an ordinary deck of 52 cards. Find the probability that a. 2 kings are drawn. b. 1 spade and 1 king other than the king of spades (in that order) are drawn. 4nPr 2 43 12 1 a. P(2 kings) 52nPr 2 52 51 2652 221 13 3 39 1 b. P(1spade1king) 52 51 2652 68 5 Examples If 2% of the auto tires manufactured by a company are defective and 2 tires are randomly selected from an entire week’s production, find the probability that neither is defective. P(neither defective ) (.98)(.98) (.98)2 .9604 Find the probability of at least 1 of the 2 selected tires is defective. P(at least 1 defective ) 1 P(none defective ) 1 .9604 0.0396 6 Example A box contains 10 computer disks and 2 are defective. If 3 disks are randomly selected From the box, find the probability that exactly 2 are defective. defective good 2nCr2 8nCr1 8 1 P(2 defective ) 10nCr3 120 15 total 7 Examples A hat contains 24 names, 13 of which are female. If seven names are randomly drawn from the hat, what is the probability that at least one male name is drawn? 13n Pr 7 1 0.9950 24n Pr 7 In a sample of 32 hand-held calculators, 26 are known to be nonfunctional. If 21 of these calculators are selected at random, what is the probability that exactly 17 in the selection are nonfunctional? Round to the nearest thousandth. 6nCr4 26nCr17 0.363 32nCr21 8 Example When printing color inserts for newspapers, it sometimes happens that the registration of the print colors is imperfect. (This results in the different colors not being aligned properly, so the image is blurry.) Suppose that in a run of 1217 one-page inserts, 78 have registration errors. If 4 inserts are chosen at random, what is the probability that at least one of them has a color registration error? (1217 78)n Pr 4 1 0.2330 1217 n Pr 4 9 Example A bag contains 5 red marbles, 6 blue marbles, 11 white marbles and 9 yellow marbles. You are asked to draw 5 marbles from the bag without replacement. What is the probability of drawing less than two yellow marbles? 22nCr5 9nCr0 22nCr4 9nCr1 P(less than 2 yellow) 31nCr5 26,334 65,835 169,911 92,169 1463 169,911 2697 END 10