Transcript Chapter 8 Notes

AP Statistics
Chapter 8 Notes
The Binomial Setting

If you roll a die 20 times, how many times
will you roll a 4? Will you always roll a 4
that many times?

The previous questions dealt with an
example of a random occurrence that
takes place in a binomial setting.
Binomial Setting
1. Each observation falls into one of just
two categories (often called “success” and
“failure”).
 2. There is a fixed number, n, of
observations.
 3. The n observations are all independent.
 4. The probability of “success”, usually
called p, is the same for each observation.

Binomial Distribution

The distribution of the count, X, of
successes in the binomial setting…

B(n, p)
– n # of observations
– p probability of success on any one
observation.
Example

In 20 rolls of a die, what is the probability
of getting exactly 3 fours?
– Why is this problem difficult to answer based
on what you have already learned?
– Is this a binomial setting?
– You can’t simply use the multiplication rule,
because the fours could be rolled in any 3 of
the 20 rolls.
Binomial Coefficient

The number of ways of arranging k successes
among n observations can be calculated by…
Read as “n choose k”
 In your calculator, n choose k can be found by
using the command nCr

Finding Binomial Probabilities
X  binomial distribution
 n  # of observations
 p  prob of success on each observation

Binomial probabilities on the
calculator
P(X = k) = binompdf (n, p, k)
 pdf  probability distribution function 

– Assigns a probability to each value of a
discrete random variable, X.
P(X < k) = binomcdf (n, p, k)
 cdf  cumulative distribution function 

– for R.V. X, the cdf calculates the sum of the
probabilities for 0, 1, 2 … up to k.
Mean and Standard Deviation
For a binomial random
variable:
 When n is large, a
binomial distribution can
be approximated by a
Normal distribution.
 We can use a Normal
distribution when.

– np > 10 and n(1 – p) > 10

If these conditions are
satisfied, then a binomial
distribution can be
approximated by…
The Geometric Setting
1. Each observation falls into one of two
categories (“success or “failure”)
 2. The observations are independent.
 3. The probability of success, p, is the
same for all observations.
 4. The variable of interest is the number
of trials required to obtain the first
success.

Calculating Geometric Probabilities

P(X = n) = (1 – p)n – 1p

“Probability that the first success occurs
on the nth trial”

P(X < n)  geometcdf (p, n)
Mean and Standard Deviation

If X is a geometric random variable with
probability of success p on each trial, then

The probability that it takes more than n
trials to the first success is…
– P(X > n) = (1 – p)n
Calculator Functions for Ch 8

Binomial
– P(X = k)  binompdf(n, p, k)
– P(X < k)  binomcdf(n, p, k)
– Simulation  randbin(n, p)

Geometric
– P(X < n)  geometcdf(p, n)

Normal
– P(min< X< max) = normalcdf(min, max, μ, σ)