S ECTION 8-6 Binomial Distribution. W ARM U P Expand each binomial ( a + b ) 2 ( x – 3 y ) 2 Evaluation each expression 4 C 3 (0.25) 0.

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Presentation transcript:

S ECTION 8-6 Binomial Distribution

W ARM U P Expand each binomial ( a + b ) 2 ( x – 3 y ) 2 Evaluation each expression 4 C 3 (0.25) 0

O BJECTIVES AND V OCABULARY Objectives Use the Binomial Theorem to expand a binomial raised to a power Find binomial probabilities and test hypotheses Vocabulary Binomial Theorem Binomial Experiment Binomial Probability

B INOMIAL D ISTRIBUTIONS You used Pascal’s Triangle to find binomial expansions in lesson 6-2. The coefficients of the expansion ( x + y ) n are the numbers in Pascal’s Triangle, which are actually combinations

B INOMIAL T HEOREM The pattern in the table can help you expand any binomial by using the Binomial Theorem

E XAMPLES : U SE THE B INOMIAL T HEOREM TO EXPAND THE BINOMIAL ( a + b ) 5 (2 x - y ) 3

B INOMIAL E XPERIMENT 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

B INOMIAL P ROBABILITY 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.

E XAMPLE Students are assigned randomly to 1 of 4 guidance counselors. What is the probability that Counselor Jenkins will get 2 of the next 3 students assigned? Ellen takes a multiple-choice quiz that has 7 questions, with 4 answer choices for each question. There is only one correct answer. What is the probability that she will get at least 2 answers correct by guessing?

E XAMPLE 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? 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?

H OMEWORK Pages #9-30, #33-37, and #39-43