2. Warm Up Expand each binomial. 1. ( a + b ) 2 2. ( x – 3 y ) 2 Evaluate each expression. 3. 4 C 3 4. (0.25) 0 5. 6. 23.2% of 37 a 2 + 2 ab + b 2 x 2 – 6 xy + 9 y 2 4 1 8.584
3. Use the Binomial Theorem to expand a binomial raised to a power. Find binomial probabilities and test hypotheses. Objectives
5. You used Pascal ’ s triangle to find binomial expansions in Lesson 6-2. The coefficients of the expansion of ( x + y ) n are the numbers in Pascal ’ s triangle, which are actually combinations.
6. The pattern in the table can help you expand any binomial by using the Binomial Theorem .
7. Example 1A: Expanding Binomials Use the Binomial Theorem to expand the binomial. ( a + b ) 5 The sum of the exponents for each term is 5. ( a + b ) 5 = 5 C 0 a 5 b 0 + 5 C 1 a 4 b 1 + 5 C 2 a 3 b 2 + 5 C 3 a 2 b 3 + 5 C 4 a 1 b 4 + 5 C 5 a 0 b 5 = 1 a 5 b 0 + 5 a 4 b 1 + 10 a 3 b 2 + 10 a 2 b 3 + 5 a 1 b 4 + 1 a 0 b 5 = a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + b 5
8. Example 1B: Expanding Binomials (2 x + y ) 3 (2 x + y ) 3 = 3 C 0 (2 x ) 3 y 0 + 3 C 1 (2 x ) 2 y 1 + 3 C 2 (2 x ) 1 y 2 + 3 C 3 (2 x ) 0 y 3 = 1 • 8 x 3 • 1 + 3 • 4 x 2 y + 3 • 2 xy 2 + 1 • 1 y 3 = 8 x 3 + 12 x 2 y + 6 xy 2 + y 3 Use the Binomial Theorem to expand the binomial.
9. In the expansion of ( x + y ) n , the powers of x decrease from n to 0 and the powers of y increase from 0 to n . Also, the sum of the exponents is n for each term. (Lesson 6-2) Remember!
10. Check It Out! Example 1a Use the Binomial Theorem to expand the binomial. ( x – y ) 5 ( x – y ) 5 = 5 C 0 x 5 (– y ) 0 + 5 C 1 x 4 (– y ) 1 + 5 C 2 x 3 (– y ) 2 + 5 C 3 x 2 (– y ) 3 + 5 C 4 x 1 (– y ) 4 + 5 C 5 x 0 (– y ) 5 = 1 x 5 (– y ) 0 + 5 x 4 (– y ) 1 + 10 x 3 (– y ) 2 + 10 x 2 (– y ) 3 + 5 x 1 (– y ) 4 + 1 x 0 (– y ) 5 = x 5 – 5 x 4 y + 10 x 3 y 2 – 10 x 2 y 3 + 5 xy 4 – y 5
11. Check It Out! Example 1b ( a + 2 b ) 3 ( a + 2 b ) 3 = 3 C 0 a 3 (2 b ) 0 + 3 C 1 a 2 (2 b ) 1 + 3 C 2 a 1 (2 b ) 2 + 3 C 3 a 0 (2 b ) 3 = 1 • a 3 • 1 + 3 • a 2 • 2 b + 3 • a • 4b 2 + 1 • 1 • 8 b 3 = a 3 + 6 a 2 b + 12 ab 2 + 8 b 3 Use the Binomial Theorem to expand the binomial.
12. A binomial experiment consists of n independent trials whose outcomes are either successes or failures; the probability of success p is the same for each trial, and the probability of failure q is the same for each trial. Because there are only two outcomes, p + q = 1, or q = 1 - p . Below are some examples of binomial experiments:
13. Suppose the probability of being left-handed is 0.1 and you want to find the probability that 2 out of 3 people will be left-handed. There are 3 C 2 ways to choose the two left-handed people: LLR, LRL, and RLL. The probability of each of these occurring is 0.1(0.1)(0.9). This leads to the following formula.
14. Example 2A: Finding Binomial Probabilities Jean usually makes half of her free throws in basketball practice. Today, she tries 3 free throws. What is the probability that Jean will make exactly 1 of her free throws? P ( r) = n C r p r q n-r P ( 1 ) = 3 C 1 (0.5) 1 (0.5) 3- 1 Substitute 3 for n, 1 for r, 0.5 for p, and 0.5 for q. = 3(0.5)(0.25) = 0.375 The probability that Jean will make exactly one free throw is 37.5%. The probability that Jean will make each free throw is , or 0.5.
15. Example 2B: Finding Binomial Probabilities Jean usually makes half of her free throws in basketball practice. Today, she tries 3 free throws. What is the probability that she will make at least 1 free throw? At least 1 free throw made is the same as exactly 1, 2, or 3 free throws made. P (1) + P ( 2 ) + P ( 3 ) 0.375 + 3 C 2 (0.5) 2 (0.5) 3- 2 + 3 C 3 (0.5) 3 (0.5) 3- 3 0.375 + 0.375 + 0.125 = 0.875 The probability that Jean will make at least one free throw is 87.5%.
16. Check It Out! Example 2a Students are assigned randomly to 1 of 3 guidance counselors. What is the probability that Counselor Jenkins will get 2 of the next 3 students assigned? The probability that Counselor Jenkins will get 2 of the next 3 students assigned is about 22%. Substitute 3 for n, 2 for r, for p, and for q. The probability that the counselor will be assigned 1 of the 3 students is .
17. Check It Out! Example 2b Ellen takes a multiple-choice quiz that has 5 questions, with 4 answer choices for each question. What is the probability that she will get at least 2 answers correct by guessing? The probability of answering a question correctly is 0.25. 5 C 2 (0.25) 2 (0.75) 5- 2 + 5 C 3 (0.25) 3 (0.75) 5- 3 + 5 C 4 (0.25) 4 (0.75) 5- 4 + 5 C 5 (0.25) 5 (0.75) 5- 5 At least 2 answers correct is the same as exactly 2, 3, 4, or 5 questions correct. P ( 2 ) + P ( 3 ) + P ( 4 ) + P ( 5 ) 0.2637 + 0.0879 + .0146 + 0.0010 0.3672
18. Example 3: Problem-Solving Application You make 4 trips to a drawbridge. There is a 1 in 5 chance that the drawbridge will be raised when you arrive. What is the probability that the bridge will be down for at least 3 of your trips?
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20. The direct way to solve the problem is to calculate P (3) + P (4). Example 3 Continued 2 Make a Plan
21. P (3) + P (4) = 4 C 3 (0.80) 3 (0.20) 4- 3 + 4 C 4 (0.80) 4 (0.20) 4- 3 = 4(0.80) 3 (0.20) + 1(0.80) 4 (1) = 0.4096 + 0.4096 = 0.8192 The probability that the bridge will be down for at least 3 of your trips is 0.8192. Example 3 Continued Solve 3
22. Look Back Example 3 Continued The answer is reasonable, as the expected number of trips the drawbridge will be down is of 4, = 3.2, which is greater than 3. 4 So the probability that the drawbridge will be down for at least 3 of your trips should be greater than
23. Check It Out! Example 3a Wendy takes a multiple-choice quiz that has 20 questions. There are 4 answer choices for each question. What is the probability that she will get at least 2 answers correct by guessing?
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25. The direct way to solve the problem is to calculate P (2) + P (3) + P (4) + … + P (20). Check It Out! Example 3a Continued An easier way is to use the complement. "Getting 0 or 1 correct" is the complement of "getting at least 2 correct." 2 Make a Plan
26. = 20 C 0 (0.25) 0 (0.75) 20- 0 + 20 C 1 (0.25) 1 (0.75) 20- 1 Check It Out! Example 3a Continued P (0) + P (1) = 1(0.25) 0 (0.75) 20 + 20(0.25) 1 (0.75) 19 0.0032 + 0.0211 0.0243 Step 1 Find P (0 or 1 correct). Step 2 Use the complement to find the probability. 1 – 0.0243 0.9757 The probability that Wendy will get at least 2 answers correct is about 0.98. Solve 3
27. Check It Out! Example 3a Continued Look Back The answer is reasonable since it is less than but close to 1. 4
28. Check It Out! Example 3b A machine has a 98% probability of producing a part within acceptable tolerance levels. The machine makes 25 parts an hour. What is the probability that there are 23 or fewer acceptable parts?
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30. The direct way to solve the problem is to calculate P(1) + P(2) + P(3) + … + P(23). Check It Out! Example 3b Continued An easier way is to use the complement. "Getting 23 or fewer" is the complement of "getting greater than 23. “ Find this probability, and then subtract the result from 1. 2 Make a Plan
31. = 25 C 24 (0.98) 24 (0.02) 25- 24 + 25 C 25 (0.98) 25 (0.02) 25- 25 Check It Out! Example 3b Continued P (24) + P (25) Step 1 Find P (24 or 25 acceptable parts). = 25(0.98) 24 (0.02) 1 + 1(0.98) 25 (0.02) 0 0.3079 + 0.6035 Step 2 Use the complement to find the probability. 1 – 0.9114 0.0886 The probability that there are 23 or fewer acceptable parts is about 0.09. 0.9114 Solve 3
32. Look Back Since there is a 98% chance that a part will be produced within acceptable tolerance levels, the probability of 0.09 that 23 or fewer acceptable parts are produced is reasonable. Check It Out! Example 3b Continued 4
33. Lesson Quiz: Part I Use the Binomial Theorem to expand each binomial. 1. ( x + 2) 4 2. (2 a – b ) 5 A binomial experiment has 4 trials, with p = 0.3. 3. What is the probability of 1 success? 4. What is the probability of at least 2 successes? x 4 + 8 x 3 + 24 x 2 + 32 x + 16 32 a 5 – 80 a 4 b + 80 a 3 b 2 – 40 a 2 b 3 + 10 ab 4 – b 5 0.4116 0.3483
34. Lesson Quiz: Part II A binomial experiment has 4 trials, with p = 0.3. 5. There is a 10% chance that Nila will have to wait for a train to pass as she heads for school. What is the probability that she will not have to wait for a train all 5 days this week? 6. Krissy has 3 arrows. The probability of her hitting the target is . What is the probability that she will get at least one arrow on the target? about 59% 78.4%
