Terms To Know. Two Types of Binomail Distribution Problems. OUTLINE 6.1 Discrete Random Variables 6.2 The Binomial Probability Distribution

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Terms To Know

OUTLINE

6.1 Discrete Random Variables

6.2 The Binomial Probability Distribution

VOCABULARY

Random variable (p. 298) Probability histogram (p. 300) Binomial random variable (p. 310) Discrete random variable (p. 298) Expected value (p. 303)

Binomial probability distribution (p. 313) Continuous random variable (p. 298) Binomial experiment (p. 310)

Cumulative distribution function (p. 316) Probability distribution (p. 299) Trial (p. 310)

MyMathLab Homework for Chapter 6

6.1 Discrete Random Variables

6.2 The Binomial Probability Distribution

For your homework, you could use StatCrunch when working on certain problems.

6.2 population size N = sample p = n = x = success: Population

Two Types of Binomail Distribution Problems

population size N = sample p = n = x = success: Population p-value known p-value unknown given random sample no random sample given 6.2

Which criteria requires that a sample size be given?

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Chapter 6 (Discrete Probability Distributions): page 2

Experiment: Roll a single fair die ten times in succession. Experiment Outcome:

Random Variable x: the number of fours facing up

: x: : x: : x:

Values of x:

Graph of x:

Experiment: A card is to selected from a well shuffled poker deck. Suppose that the casino will pay you $10 if you select

an ace. If you fail to select an ace, you are required to pay the casino $2.

Experiment Outcome: number of outcomes :

Random Variable x: gambler’s winnings for playing the game once

Values of x : Graph of x:

Experiment: A randomly selected student takes a ten-question multiple choice IQ test by guessing. Afterwards, the test is

graded.

Experiment Outcome: W C W C W W W W W W

Random Variable x: the number of correct answers

C W C C C C C C C C: x = W W W C W W W W W W: x = W C W C W W W W W W: x = W W W W W W W W W W: x = Values of x : Graph of x:

two types of outcomes

experiment outcome trial outcome

loss = negative gain

IQ TEST Name:____________ 1. Who is John?

a) John

b) none of the above 2. Who ate 9?

a) 7

b) none of the above 3. Is N.Y. city a country?

a) yes

b) none of the above 4. Is 0 a negative number?

a) yes

b) none of the above 5. What is good?

a) smoking b) none of the above 6. The color red is what color?

a) red

b) none of the above 7. What is the first integer?

a) 0

b) none of the above 8. Are footballs made from feet?

a) yes

b) none of the above 9. What is 5 divided by 0?

a) 0

b) none of the above 10. What is 0 divided by 5?

a) 0

b) none of the above from the gambler’s point of view

not from the casino’s point of view

x: x: x: experiment outcome

Random Variable

6.1 discrete number of outcomes Multiplication Rule of Counting q r s! ! ! ...

experiment outcome trial outcome number of outcomes Multiplication Rule of Counting q r s! ! !...

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Experiment: Toss a single fair coin ten times in succession. Experiment Outcome:

Random Variable x: the number of heads facing up

Values of x :

P 0

_{( )}

=

P 1

_{( )}

= P 2

_{( )}

=

P 10

_{( )}

=

cash prize: amount earned winnings: amount earned profit: amount earned - amount spent net winnings: amount earned - amount spent

Probability Distribution of a Random Variable

6.1

discrete 5 6 7 8 9 10 0 0.05 0 . 1 0.15 0 . 2 0.25 0 1 2 3 4

(the base of each rectangle is 1 unit) area=height$width

Probability Histogram 0 0.05 0 . 1 0.15 0 . 2 0.25 0 Probability Histogram

Experiment Outcome:

Values of x :

experiment outcome trial outcome number of outcomes Multiplication Rule of Counting q r s! ! ! ... 10C =2 45 P x nCx p p x n x

( )

= ! ! "

₍

1

₎

" Probability Distribution x P x

_{( )}

Probability Distribution x P x

_{( )}

P 17 1 38

( )

= P 17 1 37

( )

= 1. 2. 3. 4. ..., 36, 0, 00: P 17

_{( )}

= 1. 2. 3. 4. ..., 40, 0, 00: P 17

_{( )}

= 1. 2. 3. 4. ..., 33, 0, 00: P 17

_{( )}

= hw P# $ % & ' ( = P# $ % & ' ( =

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Experiment: A life insurance company sells a $250,000 one-year term life insurance policy to a 20-year old female for $200.

According to the National Vital Statistics Report, the probability that the female survives the year is 0.999546.

Experiment Outcome: (20-year old female policy holder)

Random Variable x: the life insurance company’s profit for a $250,000

one-year term life insurance policy for a randomly selected 20-year old female

Values of x:

Mean of a Random Variable

6.1

discrete

from the life insurance company’s point of view, not from the policy holder’s point of view

Experiment Outcome:

Values of x :

cash prize: amount earned winnings: amount earned profit: amount earned - amount spent net winnings: amount earned - amount spent

Probability Distribution

x P x

_{( )}

x P x!

_{( )}

THE BIG QUESTION

In the long run, how much will John win/lose per game?

• approximate answer using data • find exact answer using formula

1st game: $ 2nd game: $ 3rd game: $ 4th game: $ 5th game: $ 6th game: $ . . . . . . x x n =

)

%x=

)

*+x P x!

( )

,-The insurance company is gambling on whether the 20-year old female policy holder will die before the policy expires in one year.

Probability Distribution x P x

_{( )}

x P x!

_{( )}

%x: Mean or Expected Value %x=

)

*+x P x!

( )

,-%x=

)

*+x P x!

( )

,-: Mean %x: Mean or Expected Value

If the insurance company sells many policies, the company is expected to $ per policy, on average. make/lose

If John plays many games, he is expected to $ per game, on average. win/lose

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Standard Deviation of a Random Variable

6.1 discrete .x x P x %x 2 2 2 =

)

*+ !

_{( )}

,-" .x x %x P x 2 2 =

₎

*₊

₍

"

₎

!

_{( )}

, -.x x %x P x 2 2 =

)

*₊

₍

"

₎

!

_{( )}

,_-: Variance .x x P x %x 2 2 2 =

)

*+ !

_{( )}

,-" : Variance .x= .x 2 _{: Standard Deviation} Probability Distribution %x: .x 2_: _- ₌ .x:

3 decimal places 3 decimal places 3 decimal places 3 decimal places 1 decimal place .x= .x 2 .x= .x 2 1 decimal place .x: fill in x P x

_{( )}

x P x!

_{( )}

x"%_x

₍

x"%_x

₎

2

₍

x"%_x

₎

2!P x

_{( )}

x P x

_{( )}

x P x!

_{( )}

x2 _x2_!_{P x}

( )

fill in

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Factorial and Combination Formulas

6.2

n!= ! ! ! !1 2 3 ... n: factorial formula nCr n r n r = !

₍

"

₎

! ! !: combination formula n!= ! ! ! !1 2 3 ... n 1!: 2!: 3!: 4!: 5!: 6!: 7!: 8!: 0!: nCr n r n r = !

₍

"

₎

! ! ! 3 1 3 1 3 1 C = !

₍

"

₎

! ! !: 7 0 7 0 7 0 C = !

₍

"

₎

! ! !: 5 2 5 2 5 2 C = !

₍

"

₎

! ! !: 8 5 8 5 8 5 C = !

₍

"

₎

! ! !: n factorial n choose r n! nCr nCr n r = # $ % & ' ( same thing

Only Use Calculator

20C12: 25C15: 40C30:

20 12

MATH PRB 3:_n _r ENTER ENTER

n r

C

C =

22 15

22 115

MATH PRB 3:n r ENTER ENTER

n r

C

C =

7 7

MATH PRB 4:! ENTER ENTER x!

8 8

MATH PRB 4:! ENTER ENTER x!

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Binomial Distribution

6.2 Experiment: Roll a single fair die two times in succession.

Experiment Outcome: Trial Outcome: Number of trials: Success for a trial:

Failure for a trial: , , , ,

Probability of success for a trial: p =

Probability of failure for a trial: 1" p =

Random Variable x: number of successes

x = x =

Values of x :

Binomial Probability Function P x

_{( )}

: P x C_x

x x

( )

= !# $ % & ' ( !# $ % & ' ( " 2 2 1 6 5 6 P 0 C 1 6 5 6 2 0 0 2 0

( )

= !# $ % & ' ( !# $ % & ' ( " : 2C0: P 1 C 1 6 5 6 2 1 1 2 1

( )

= !# $ % & ' ( !# $ % & ' ( " : 2C1: P 2 C 1 6 5 6 2 2 2 2 2

( )

= !# $ % & ' ( !# $ % & ' ( " : 2C2:

equally likely outcomes

experiment outcome trial outcome

Binomial Random Variable

x: number of successes out of n

Binomial Probability Function

P x nCx p p x n x

( )

= ! ! "

₍

1

₎

" for 0 / /x n 4 decimal places 4 decimal places 4 decimal places x P x

_{( )}

x P x!

_{( )}

x"%_x

₍

x"%_x

₎

2

₍

x"%_x

₎

2!P x

_{( )}

Binomial Distribution Shortcut Formulas %x=np ._x2 np ₁ p =

₍

"

₎

.x= np

(

1"p

)

%x= # $ % & ' ( 2 1 6 : .x= # $ % & ' (# $ % & ' ( 2 1 6 5 6 : 4 decimal places 4 decimal places Discrete Distribution General Formulas %x=

)

*+x P x!

( )

,-: .x=

)

*₊

(

x"%x

)

!P x

( )

, -2 : 4 decimal places

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Experiment: A random sample of 5 light bulbs is taken from a shipment of 1,000,000 light bulbs

with a 1% defect rate. Sampling is done without replacement.

Experiment Outcome: Trial Outcome: Number of trials:

Success for a trial: defective bulb Failure for a trial: non-defective bulb Probability of success for a trial: p = Probability of failure for a trial: 1" p =

Random Variable x: number of successes

x = x =

Values of x :

Binomial Probability Function P x

_{( )}

: P x

_{( )}

=5C_x!

₍

₎

x!

₍

₎

"x

5 0 01. 0 99. P 0 5C0 0 01 0 99 0 5 0

( )

= !

₍

.

₎

!

₍

.

₎

" : 5C0: P 1 5C1 0 01 0 99 1 5 1

( )

= !

₍

.

₎

!

₍

.

₎

" : 5C1: P

_{( )}

2 =₅C₂!

₍

0 01.

₎

2!

₍

0 99.

₎

5 2" : 5C2: P

_{( )}

3 =₅C₃!

₍

0 01.

₎

3!

₍

0 99.

₎

5 3" : $ 10 5C3: P

_{( )}

4 =₅C₄!

₍

0 01.

₎

4!

₍

0 99.

₎

5 4" : $ 10 5C4: P 5 5C5 0 01 0 99 5 5 5

( )

= !

₍

.

₎

!

₍

.

₎

" : $ 10 5C5:

Binomial Distribution

6.2

Binomial Random Variable

x: number of successes out of n

Binomial Probability Function

P x _nC_x px p n x

( )

= ! ! "

₍

1

₎

" for 0 / /x n x P x

_{( )}

x P x!

_{( )}

x"%x

(

x"%x

)

2 x" x P x

(

%

)

2!

_{( )}

Binomial Distribution Shortcut Formulas %x=np .x np p 2 ₁ =

₍

"

₎

.x= np

(

1"p

)

%x=5 0 01

(

.

)

: .x= 5 0 01 0 99

(

.

)(

.

)

: 4 decimal places 2 decimal places 4 decimal places 4 decimal places 4 decimal places scientific notation scientific notation scientific notation Discrete Distribution General Formulas %x=

)

*+x P x!

( )

,-: .x=

)

*₊

(

x"%x

)

!P x

( )

, -2 : 4 decimal places 4 decimal places

N

=

large/small sampling without replacement

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Binomial Distribution

6.2

only %x=

)

*+x P x!

( )

,-only %x=np

both %x=

)

*+x P x!

( )

,- and %x=np

none of the above

only .x=

)

*₊

(

x"%x

)

!P x

( )

, -2 only .x= np

(

1"p

)

both .x=

)

*₊

(

x"%x

)

!P x

( )

, -2 and .x= np

(

1"p

)

none of the above

N

=

n

=

N

=

n

=

p : probability of success for a trial or proportion of the population being successful % of the population at home.

large/small

sampling without replacement

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P< % P< %

Binomial Distribution

6.2

Binomial Distribution: n = 300, p = 0.25

np

₍

1"p

₎

:

Is the binomial histogram bell-shaped? %: . :

Which x-values are considered unusual? whole number

yes/no

2 decimal places 2 decimal places

Binomial Distribution: n = 200, p = 0.90 np

₍

1"p

₎

:

Is the binomial histogram bell-shaped? %: . :

Which x-values are considered unusual? whole number

yes/no

2 decimal places 2 decimal places

whole number whole number whole number

unusual result unusual result

0

whole number whole number whole number

unusual result unusual result 0 200x x x = 0, 1, 2, 3, ..., 300 x = 0, 1, 2, 3, ..., 200 %x=np .x= np

(

1"p

)

%x=np .x= np

(

1"p

)

unusually event: P < see pages 225-226