AP Stats unit 4Binomial and Geometric Distributions

The Binomial Setting

1. Two outcomes: "success" and "failure"

2. Independent Outcomes: knowing the results of one observation tells you nothing about the other observations (card is shuffled back in)

3. Same probability for each success (p)

4. Fixed number of trials (n)

Binomial Probability Formula

P(x=k): C(n,k) (p)^k (q)^(n-k)

(given on formula sheet)

binompdf(7, 0.7, 5): C(7,5)(0.7)^5(0.3)2

B(n,p)

When a distribution is binomial:

n= number of observations/trials

p= probability of success

-- B(6, 0.6)

binompdf(n,p,k)

Binomial Distributions on the Calculator

- Found under "DISTR" (2nd "VARS"), "A:" (below "0:")

n= fixed number of observations/trials

p= probability of success

k= success number of interest

Same result as equation/formula

What is the probability that there are two successes (k=2)

binomcdf(n,p,k)

Finds the probability of k and each values from 0 to k (includes k)

-- c alludes to this function being "cumulative" (whereas the p in pdf can be seen as "picky")

- Found under the pdf function

Use with words like: "at least "or fewer" "or more" "at most"

- the statement must be "or equal to"

- statement must go up from zero-- "less than"

-- may need to do 1 - binomcdf to get desired outcome

"fewer than 3 kids": P(x<3) --> P(x/4) --> 1 - P(x<4) --> 1 - P(x/1) --> 1 - P(x=0)

Binomial Chart

- Pick Ks of varying values

- find P(x=k) for each -- binompdf(n,p,k)

- find the cumulative as k increases (should add up to one once k=n)

--> to find binomcdf without a calculator, add up all the binompdfs for all the values leading up to k, including k

Binomial Distribution Mean

u = np

(given on formula sheet)

the expected values of successes

Binomial Distribution standard deviation

sqroot(npq)

-- q = 1-p

(given on formula sheet)

Can the binomial distribution be normally approximated?

IF

np >/ 10 and n(p-1)>/10

THEN

N(mu, sigma) or N(np, sqroot(npq))

Normal Approximations

- Draw a picture

- Find the z scores ( #-mean / standard deviation)

- Find on chart (always gives values of flag pointing to start/left)

- Do any final math calculations

Using calculator binom is more accurate

binomia

1. Two outcomes: "success" and "failure"

2. Independent Outcomes: knowing the results of one observation tells you nothing about the other observations (card is shuffled back in)

3. Same probability for each success (p)

4. Don't know how many trials

Can't use normal approximations

Difference between a Binomial and Geometric Distribution

The variable of interest for a binomial distribution is the number of success that occur (X)

The variable of interest for a geometric distribution is the number of trials it takes to obtain the first success (X)

Geometric Probability Formula

P(x=n): (1-p)^(n-1) p

- probability of it failing every time before n, and probability that it succeeds once

--> still must be independent (must put the card back in the deck)

geometpdf(p,n)

P(x=n)

p = probability of success

n= trial number of interest

The probability that it takes n tried for a success to occur

geometcdf(p,n)

P(x P(x>n) --> 1 - binomcdf(p,n)

Geometric Distribution Mean

u = 1/p

the expected number of trials for a success to occur

Geometric Distribution Standard Deviation

sqroot( (1-p)/(p^2) )

Interpret Binomial mean

*21% of cell phone owners have a cracked screen. If the BTB team randomly sleects 40 people who own cell phones, what is the expected number with cracked screens?

*calcualte means with equation= 8.4 cell phones with cracked screens. In many random samples of 40 cell phone owners, the BTB team can expect an average of 8.4 people to have a cracked cell phone screen

Interpret Binomial standard deviation- same example

*In many random samples of 40 cell phone owners, the number with cracked phones will typically vary from the mean of 8.4 by about 2.58 phones

Binomial conditions

BINS

B- Binary success or failure

I- Independent trails

N= Number of trails is fixed

S- same probability of success

Geometric conditions

BITS

B- binary success or failure

I- independent trials

T- trials continue until first success

S- same probability of success