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warm­up

Skills we need

Skills we've learned

8­6 Binomial Distributions

Find the following probabilities when rolling 3 dice:1) P(exactly 1 three) 2) P(exactly 2 threes)3) P(at least 1 three)

4) Expand

Warm­up ans.

Warm­up AnswersFind the following probabilities when rolling 3 dice:1) P(exactly 1 three) 2) P(exactly 2 threes)3) P(at least 1 three)

4) Expand

P(1) =

P(2) =

P(1) + P(2) + P(3) =

++ =

Or... 1 P(0) =–

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Learning Target

8­6 Binomial Distributions

1. Use the Binomial Theorem to expand a binomial raised to a power.

2. Find binomial probabilities and test hypotheses.

Determining your chances of winning a marketing contest.

Binomial Theorem

I. Binomial Theorem

1. Expand *see warm­up

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You Try

2. Expand

Binomial Experiment

II. 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.

3) P(exactly 2 heads out of 10 flips) =

P(r) = nCr prqn­r p = 0.5 and q = 0.5P(2) = 10C2 (0.5)2(0.5)10­2

=

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You try

4) Ellen takes a multiple­choice 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?

P(r) = nCr prqn­r

P(at least 2) = P(2) + P(3) + P(4) + P(5)

Or P(at least 2) = 1 [P(0) + P(1)]–

5C0 (0.25)0(0.75)5­0 5C1 (0.25)1(0.75)5­11 [ + ]–1 [.23730468 + .3955078] ≈ .367–

You try

5) 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 probability that the bridge will be down for at least 3 of your trips is 0.8192.

P(3) + P(4) =

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6) 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?

1 [P(25) + P(24)] =–

So

Let's think

Can the binomial theorem find the probability of drawing exactly 2 hearts in a hand of 5 cards?

Pepys wrote a letter to Isaac Newton in 1693 asking about a wager he planned to make. Pepys asked which was more likely: 1. at least one 6 when 6 dice are rolled2. at least two 6's when 12 dice are rolled3. at least three 6's when 18 dice are rolled

see #47 of your homework for this answer

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Homework

8.6 p.590 #2, 6, 8, 10, 14, 21, 25, 26 ­ 29, 32, 36, 41 ­ 43