Chapter 8 Binomial and Geometric Distributions
The binomial setting consists of the following 4
characteristics:
1) Each observation falls into one of two categories –
“success” or “failure”
2) There is a fixed number of observations n
3) The n observations are all independent
4) The probability of success, p, is the same for each
observation
•The distribution of the number of successes in the
binomial setting is the binomial distribution with
parameters n and p.
•n represents the number of observations.
•p represents the probability of success on any one
observation.
•The possible values of X are the whole numbers from 0 to
n.
•As an abbreviation, we say B(n, p).
1
Do the following situations describe the binomial setting?
If so, describe their distribution.
1) A child has 0.25 chance of getting type O blood if both
of his parents are carriers for type O blood. Let X =
the number of children among these parent’s 5 children
that have type O blood.
2) Deal 10 cards from a deck of cards and count the
number of red cards.
3) Choose an SRS of 10 switches from a shipment of
10000 switches (of which 10% are bad) Let X = the #
of bad switches in the SRS.
Choose an SRS of 10 switches from a shipment of 10,000
switches (of which 10% are bad) Let X = the # of bad
switches in the SRS.
What is the probability that no more than 1 of the switches
in the sample fail inspection?
P( X ≤ 1) = P( X = 0) + P( X + 1)
2
The Formulas!!!
Binomial Pdf is found by computing the formula:
n
n−k
P ( X = k ) =   p k (1 − p )
k 
Where
n
 
k 
is equal to:
n!
k !( n − k )!
n
  Represents the number of ways of arranging k successes
 k  among n observations.
Ex: Find the probability of getting 5 correct on the
multiple choice “test” we did yesterday.
b) Now find the probability of getting 5 or fewer correct.
3
If a basketball player makes 25% of her free throws, what is
the mean number you would expect her to make in 12 free
throws?
This leads us to the mean and standard deviation of a
binomial random variable.
µ = np
σ = np (1 − p )
Ex: Find the mean and standard deviation of the number of
bad switches is an SRS of 10 switches with the information
we used before.
4
Using your calculator to compute binomial probabilities:
Corrine is a basketball player who makes 75% of her free
throws over the course of a season. In a key game she
shoots 12 free throws and only makes 7 of them. Is it
unusual for her to make 7 of 12 free throws if she is really
a 75% free throw shooter?
You can use your calculator to compute the binomial
probability by entering binompdf(n, p, X).
While the pdf command calculates one probability, the cdf
command calculates the sum of the probabilities up to (and
including) X.
The command for cdf is binomcdf(n, p, X)
Let’s revisit the example with Corinne and her free throws.
Corrine is a basketball player who makes 75% of her free throws over the
course of a season. In a key game she shoots 12 free throws and only
makes 7 of them. The fans think she choked because she was nervous. Is it
unusual for Corinne to perform this poorly?
5
Just as with most distributions we discuss, we want to be
able to use a normal distribution to describe a binomial
setting.
When n is large, we can use normal probability
calculations to approximate binomial probabilities.
How large is large? As a rule of thumb, we will use:
np ≥ 10 and n(1 − p) ≥ 10
So we will say when n is large, the distribution of X is
approximately normal N np, np (1 − p )
(
)
Ex: Are attitudes towards shopping changing? Sample
surveys show that fewer people are shopping than in the
past. 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 US residents
would say agree if asked the same question. What is the
probability that 1520 or more of the sample agree?
6
Simulating Binomial Experiments
You can use the randbin key on your calculator to simulate
binomial situations. If we look at the example with our
basketball player making 75% of her shots and shooting 12
shots, we can run a simulation to see how many shots out of
12 she would make on average or to see how many times she
would make 7 or fewer shots.
7