Binomial Distributions

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11-6
Binomial Distributions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Expand each binomial. 1. (a b)2 2.
(x 3y)2 Evaluate each expression. 3.
4C3 4. (0.25)0 5. 6. 23.2 of 37
x2 6xy 9y2
a2 2ab b2
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4
8.584
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Objectives
Use the Binomial Theorem to expand a binomial
raised to a power. Find binomial probabilities
and test hypotheses.
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Vocabulary
Binomial Theorem binomial experiment binomial
probability
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You used Pascals triangle to find binomial
expansions in Lesson 6-2. The coefficients of the
expansion of (x y)n are the numbers in Pascals
triangle, which are actually combinations.
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The pattern in the table can help you expand any
binomial by using the Binomial Theorem.
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Example 1A Expanding Binomials
Use the Binomial Theorem to expand the binomial.
(a b)5
The sum of the exponents for each term is 5.
(a b)5 5C0a5b0 5C1a4b1 5C2a3b2 5C3a2b3
5C4a1b4 5C5a0b5
1a5b0 5a4b1 10a3b2 10a2b3 5a1b4
1a0b5
a5 5a4b 10a3b2 10a2b3 5ab4 b5
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Example 1B Expanding Binomials
Use the Binomial Theorem to expand the binomial.
(2x y)3
(2x y)3 3C0(2x)3y0 3C1(2x)2y1 3C2(2x)1y2
3C3(2x)0y3
1 8x3 1 3 4x2y 3 2xy2 1 1y3
8x3 12x2y 6xy2 y3
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Check It Out! Example 1a
Use the Binomial Theorem to expand the binomial.
(x y)5
x5 5x4y 10x3y2 10x2y3 5xy4 y5
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Check It Out! Example 1b
Use the Binomial Theorem to expand the binomial.
(a 2b)3
a3 6a2b 12ab2 8b3
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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
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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.
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Example 2A Finding Binomial Probabilities
Jean usually makes half of her free throws in
basketball practice. Today, she tries 3 free
throws. What is the probability that Jean will
make exactly 1 of her free throws?
Substitute 3 for n, 1 for r, 0.5 for p, and 0.5
for q.
P(r) nCrprqn-r
P(1) 3C1(0.5)1(0.5)3-1
3(0.5)(0.25) 0.375
The probability that Jean will make exactly one
free throw is 37.5.
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Example 2B Finding Binomial Probabilities
Jean usually makes half of her free throws in
basketball practice. Today, she tries 3 free
throws. What is the probability that she will
make at least 1 free throw?
At least 1 free throw made is the same as exactly
1, 2, or 3 free throws made.
P(1) P(2) P(3)
0.375 3C2(0.5)2(0.5)3-2 3C3(0.5)3(0.5)3-3
0.375 0.375 0.125 0.875
The probability that Jean will make at least one
free throw is 87.5.
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Check It Out! Example 2a
Students are assigned randomly to 1 of 3 guidance
counselors. What is the probability that
Counselor Jenkins will get 2 of the next 3
students assigned?
The probability that Counselor Jenkins will get 2
of the next 3 students assigned is about 22.
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Check It Out! Example 2b
Ellen takes a multiple-choice 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?
0.2637 0.0879 .0146 0.0010 ? 0.3672
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Example 3 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?
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Example 3 Continued
The answer will be the probability that the
bridge is down at least 3 times.
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Example 3 Continued
The direct way to solve the problem is to
calculate P(3) P(4).
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Example 3 Continued
P(3) P(4)
4C3(0.80)3(0.20)4-3 4C4(0.80)4(0.20)4-3
4(0.80)3(0.20) 1(0.80)4(1)
0.4096 0.4096
0.8192
The probability that the bridge will be down for
at least 3 of your trips is 0.8192.
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Example 3 Continued
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.
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Check It Out! Example 3a
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 probability that Wendy will get at least 2
answers correct is about 0.98.
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Check It Out! Example 3b
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 probability that there are 23 or fewer
acceptable parts is about 0.09.