Section 8 6
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Section 8-6. Binomial Distribution. Warm Up. Expand each binomial ( a + b ) 2 ( x – 3 y ) 2 Evaluation each expression 4 C 3 (0.25) 0. Objectives and Vocabulary. Objectives Use the Binomial Theorem to expand a binomial raised to a power

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Section 8-6

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Section 8 6

Section 8-6

Binomial Distribution


Warm up

Warm Up

  • Expand each binomial

    • (a + b)2

    • (x – 3y)2

  • Evaluation each expression

    • 4C3

    • (0.25)0


Objectives and vocabulary

Objectives and Vocabulary

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


Binomial distributions

Binomial Distributions

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


Binomial theorem

Binomial Theorem

  • The pattern in the table can help you expand any binomial by using the Binomial Theorem


Examples use the binomial theorem to expand the binomial

Examples: Use the Binomial Theorem to expand the binomial

  • (a + b)5

  • (2x- y)3


Binomial experiment

Binomial Experiment

  • 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


Binomial probability

Binomial Probability

  • 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 3C2 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.


Example

Example

  • 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?


Example1

Example

  • 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?


Homework

Homework

  • Pages 590-593 #9-30, #33-37, and #39-43


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