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Test 4 and 5Wednesday Oct 30th

guessing

Suppose there is multiple choice quiz on a subject you don’t know anything about…. 15th Century Russian Literature; Nuclear physics etc.

You have to guess on every question. There are 5 questions and each question

has 4 choices.

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Let x be the score on the test.Find p(x=0)

In another words the probability you will get a score of zero, i.e. you will get all the questions wrong Find p(x=1)

In another words the probability you will get a score of 1, i.e. you will get only one question correct.

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Question number

Correct or wrong

1

2

3

4

5

Repeat the process:

P(2)=

P(3)=

p(4)=

P(5)=

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What if the number of questions changed Let’s say now the test has 10 questions

and each question has 4 choices. What does the probability distribution chart

looks like?

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x P(x)

012345678910

What if the number of choices changes Let’s say now the test has 10 questions

and each question has 5 choices. What does the probability distribution chart

looks like?

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12345678910

5-3 The Binomial Distribution

10

Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes.

Examples include: when a coin is tossed it can land on heads or

tails,

when a baby is born it is either a boy or girl.

It will rain or it won’t

A person will pass the bar exam or not.

The Binomial Distribution

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The binomial experiment is a probability experiment that satisfies these requirements:

1. Each trial can have only two possible outcomes—success or failure.

2. There must be a fixed number of trials.

3. The outcomes of each trial must be independent of each other.

4. The probability of success must remain the same for each trial.

Notation for the Binomial Distribution

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The symbol for the probability of success

The symbol for the probability of failure

The numerical probability of success

The numerical probability of failure

and P(F) = 1 – p = q

The number of trials

The number of successes

P(S)

P(F)

p

q

P(S) = p

n

X

Note that X = 0, 1, 2, 3,...,n

The Binomial Distribution

!

- ! ! X n Xn

P X p qn X X

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In a binomial experiment, the probability of exactly X successes in n trials is

number of possible probability of adesired outcomes desired outcome

or

X n Xn xP X C p q

Chapter 5Discrete Probability Distributions

Section 5-3Example 5-16

Page #272

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Example 5-16: Survey on Doctor Visits

A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month.

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!

- ! ! X n Xn

P X p qn X X

3 7

10! 1 43

7!3! 5 5

P

1510,"one out of five" , 3 n p X

0.201

Chapter 5Discrete Probability Distributions

Section 5-3Example 5-17

Page #273

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Example 5-17: Survey on EmploymentA survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs.

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3 25!3 0.30 0.70

2!3! P

5, 0.30,"at least 3" 3,4,5 n p X

0.132

4 15!4 0.30 0.70

1!4! P 0.028

5 05!5 0.30 0.70

0!5! P 0.002

3 0.132

0.028

0.002

0.162

P X

Chapter 5Discrete Probability Distributions

Section 5-3Example 5-18

Page #273

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Example 5-18: Tossing CoinsA coin is tossed 3 times. Find the probability of getting exactly two heads, using Table B.

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123, 0.5, 2 n p X 2 0.375 P

The Binomial Distribution

Mean: np2Variance: npq

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The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas.

Standard Deviation: npq

Chapter 5Discrete Probability Distributions

Section 5-3Example 5-23

Page #276

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Example 5-23: Likelihood of TwinsThe Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins.

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8000 0.02 160 np

2 8000 0.02 0.98 156.8 157 npq

8000 0.02 0.98 12.5 13 npq

Tech notes

Read technology notes on page 281.

Read example 5-19 on page 274

Exercises 5.3

Page 276 #1, 5, 11, 15 and 17

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