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### Section 8-6

Binomial Distribution

Warm Up

- Expand each binomial
- (a + b)2
- (x – 3y)2

- Evaluation each expression
- 4C3
- (0.25)0

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

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

- 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

- (a + b)5
- (2x- y)3

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

- 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

- 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?

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

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

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