Concepts Examples Combinations Formula If C1n, r2 represents the number of combinations of n elements taken r at a time, with r … n, then the following holds true. C1n, r2 = n! 1 n −r2! r! , or C1n, r2 = n! r!1n −r2! How many committees of 4 senators can be formed from a group of 9 senators? The arrangement of senators does not matter, so this is a combinations problem. C19, 42 = 9! 4!19 - 42! = 126 committees Let n = 9 and r = 4. A number is chosen at random from S = 51, 2, 3, 4, 5, 66. What is the probability that the number is less than 3? The event is E = 51, 26, n1S2 = 6, and n1E2 = 2. P1E2 = 2 6 = 1 3 What is the probability that the number is 3 or more? This event is E′. P1E′2 = 1 - 1 3 = 2 3 An experiment consists of rolling a fair die 8 times. Find the probability that exactly 5 rolls result in a 2. P15 twos in 8 rolls2 = a 8 5b a 1 6b 5a5 6b 8-5 Let n = 8, r = 5, p = 1 6 , and q = 5 6 . = 56 a 1 6b 5a5 6b 3 ≈0.004 Write the first five terms of each sequence. Determine whether the sequence is arithmetic, geometric, or neither. 1. an = n n + 1 2. an = 1-22n 3. a n = 21n + 32 4. an = n1n + 12 5. a1 = 5 an = an-1 - 3, if n Ú 2 6. a1 = 1, a2 = 3, an = an-2 + an-1, if n Ú 3 7. Concept Check Write an arithmetic sequence that consists of five terms, with first term 4, having the sum of the five terms equal to 25. Write the first five terms of each sequence described. 8. arithmetic; a2 = 10, d = -2 9. arithmetic; a3 = p, a4 = 1 10. geometric; a1 = 6, r = 2 11. geometric; a1 = -5, a2 = -1 Chapter 11 Review Exercises 1115 CHAPTER 11 Review Exercises 11.7 Basics of Probability Probability of Event E In a sample space S with equally likely outcomes, the probability of event E is determined as follows. P1E2 = n1E2 n1S2 Properties of Probability For any events E and F, the following hold true. 1. 0 … P1E2 … 1 2. P1a certain event2 = 1 3. P1an impossible event2=0 4. P1E′2 = 1 - P1E2 5. P1E or F2 = P1E´F2 = P1E2 + P1F2 - P1E¨F2 Binomial Probability Let p represent the probability of a success, and let q = 1 - p represent the probability of a failure. In a binomial experiment, the probability of obtaining exactly r successes in n trials is found as follows. P1r successes in n trials2 = a n r b p rqn−r
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