Section 8.1 Binomial Distributions AP Statistics.

Section 8.1Binomial Distributions

AP Statistics

AP Statistics, Section 8.1.1 2

The Binomial Setting1. Each observation falls into one of just

two categories, which for convenience we call “success” or “failure”

2. There are a fixed number n of observations

3. The n observations are all independent.4. The probability of success, call it p, is the

same for each observation.


The Binomial Setting: Example1. Each observation falls into one of just two

categories, which for convenience we call “success” or “failure”: Basketball player at the free throw.

2. There are a fixed number n of observations: The player is given 5 tries.

3. The n observations are all independent: When the player makes (or misses) it does not change the probability of making the next shot.

4. The probability of success, call it p, is the same for each observation: The player has an 85% chance of making the shot; p=.85


Shorthand Normal distributions can be described using the

N(µ,σ) notation; for example, N(65.5,2.5) is a normal distribution with mean 65.5 and standard deviation 2.5.

Binomial distributions can be described using the B(n,p) notation For example, B(5, .85) describes a binomial distribution with 5

trials and .85 probability of success for each trial.


Example Blood type is inherited. If both parents carry

genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations.

How would you describe this with “B” notation? X=B( )


Example Deal 10 cards from a shuffled deck and count

the number “X” of red cards. A “success” is a red card.

How would you describe this using “B” notation?

This is not a Binomial distribution because once you pull one card out, the probabilities change.


Example Blood type is inherited. If both parents carry genes for

the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations.

What is the probability that 3 children are type O?


Binomial Coefficient

Sometimes referred to as “n choose k”

For example: “I have 10 students in a class. I need to choose 2 of them.”

In these examples, order is not important.

!

! !

10 10!

2 2! 10 2 !

10 9 8 7 6 5 4 3 2 1

2 1 8 7 6 5 4 3 2 1

45

n n

k k n k


Binomial Coefficients on the Calculator


Binomial Probabilities

( ) 1n kkn

P X k p pk


Binomial Distributions on the calculator Corinne makes 75% of her

free throws. What is the probability of

making exactly 7 of 12 free throws.

Binomial Probabilities B(n,p) with k successes binompdf(n,p,k) binompdf(12,.75,7)=.1032

1n kkn

p pk

5712.75 .25

7


Corinne makes 75% of her free throws.

What is the probability of making at most 7 of 12 free throws.

B(n,p) with k successes binomcdf(n,p,k) binomcdf(12,.75,7)=.1576

57 6 6

5 7 4 8

3 9 2 10

1 11 0 12

12 12.75 .25 .75 .25

7 6

12 12.75 .25 .75 .25

5 4

12 12.75 .25 .75 .25

3 2

12 12.75 .25 .75 .25

1 0

Binomial Distributions on the calculator


Corinne makes 75% of her free throws.

What is the probability of making at least 7 of 12 free throws.

B(n,p) with k successes binomcdf(n,p,k) 1-binomcdf(12,.75,6)=

57 8 4

9 3 10 2

11 1 12 0

12 12.75 .25 .75 .25

7 8

12 12.75 .25 .75 .25

9 10

12 12.75 .25 .75 .25

11 12


Mean and Standard Deviation of a Binomial Distribution

1

np

np p


(10,.5), (5, 10*.5*.5)B N


(100,.5), (50, 100*.5*.5)B N


(1000,.5), (500, 1000*.5*.5)B N


Example: A recent survey asked a nationwide random

sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time-consuming.” Suppose that in fact 60% of all adults would “agree”. What is the probability that 1520 or more of the sample “agree”.


Normal Approximation of Binomial Distribution As the number of trials n gets larger, the

binomial distribution gets close to a normal distribution.

Question: What value of n is big enough? The book does not say, so let’s see how the close two calculations are…


Calculator B(2500,.6) or P(X>1520) 1-binomcdf(2500,.6,1519) .2131390887

nCDF(1520, 1E99, 1500, 24.495) P(X>1520)=.207


Normal Approximations for Binomial Distributions

As a “rule of thumb,” we may use the Normal Approximation when…

np ≥ 10 and n (1 – p) ≥ 10


Here are some useful applications of the binomcdf and binomcdf commands:

To find P(x = k), use binompdf(n,p,k) To find P(x ≤ k), use binomcdf(n,p,k) To find P(x < k), use binomcdf(n,p,k-1) To find P(x > k), use 1-binomcdf(n,p,k) To find P(x ≥ k), use 1-binomcdf(n,p,k-1)


Homework

Binomial Worksheet

Section 8.1 Binomial Distributions AP Statistics.

Documents

Transcript of Section 8.1 Binomial Distributions AP Statistics.