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Learn how to expand binomials using the Binomial Theorem and solve probability problems using binomial probabilities and hypotheses testing.
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Essential Questions • How do we use the Binomial Theorem to expand a binomial raised to a power? • How do we find binomial probabilities and test hypotheses?
1 3 Solve Make a Plan Understand the Problem 4 2 Problem-Solving Application 4 Steps for Problem Solving Look Back
1 • • The probability that the drawbridge will be down is = 0.8. Understand the Problem Example 1: Problem-Solving Application 1 in 5 chance 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? probability that the bridge will be down will be raised The answer will be the probability that the bridge is down at least 3 times. List the important information: • • You make 4 trips to the drawbridge.
Make a Plan 2 Example 1: 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? The direct way to solve the problem is to calculate P(3) + P(4).
3 Solve Example 1: 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? P(3) + P(4) = 4C3 (0.80)3 (0.20)1 + 4C4 (0.20)0 (0.80)4 The probability that the bridge will be down for at least 3 of your trips is 0.8192.
So the probability that the drawbridge will be down for at least 3 of your trips should be greater than 4 Example 1: 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? Look Back 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.
1 • • The probability of guessing a correct answer is . Understand the Problem Example 2: Problem-Solving Application 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? The answer will be the probability she will get at least 2 answers correct by guessing. List the important information: • • Twenty questions with four choices
Make a Plan 2 Example 2: Problem-Solving Application 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? The direct way to solve the problem is to calculate P(2) + P(3) + P(4) + … + P(20). An easier way is to use the complement. "Getting 0 or 1 correct" is the complement of "getting at least 2 correct."
3 Solve Example 2: Problem-Solving Application 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? Step 1 Find P(0 or 1 correct). P(0) + P(1) = 20C0 (0.25)0 + 20C1 (0.75)19 (0.75)20 (0.25)1 Step 2 Use the complement to find the probability. The probability that Wendy will get at least 2 answers correct is about 0.98.
4 Example 2: Problem-Solving Application 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? Look Back The answer is reasonable since it is less than but close to 1.
1 Understand the Problem Example 3: Problem-Solving Application 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? The answer will be the probability of getting 1–23 acceptable parts. • List the important information: • • 98% probability of an acceptable part • • 25 parts per hour with 1–23 acceptable parts
Make a Plan 2 Example 3: Problem-Solving Application 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? The direct way to solve the problem is to calculate P(1) + P(2) + P(3) + … + P(23). 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.
3 Solve Example 3: Problem-Solving Application 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? Step 1 Find P(24 or 25 acceptable parts). P(24) + P(25) (0.02)1 = 25C24 (0.98)24 + 25C25 (0.98)25 (0.02)0 Step 2 Use the complement to find the probability. The probability that there are 23 or fewer acceptable parts is about 0.09.
4 Example 3: Problem-Solving Application 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? 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.
