Download The Binomial Distribution

The Binomial
Distribution
Chapter 7 Section 3
When a probability problem can be reduced
to two outcomes or only has two
outcomes, it is called a binomial
experiment. It must also meet the
following requirements:
a.) must be a fixed number of trials
b.) outcomes of trials must be independent
c.) probability for a success must remain
the same for each trial.
Binomial Experiment
The outcomes of a binomial experiment and the
corresponding outcomes are called a binomial
distribution.
Notation for Binomial Distribution:
P(S)
symbol for the probability of success
P(F)
symbol for the probability of failure
p
numerical probability of success
q
numerical probability of failure
P(S) = p
P(F) = 1 – p = q
n
number of trials
X
number of successes in n trials
Binomial Distribution
In a binomial experiment, the probability of
exactly X successes in n trials is:
n!
X
n X
P(X) 
 p q
(n  X)!X!
Binomial Probability Formula
A coin is tossed 3 times. Find the
probability of getting exactly two heads.
This is a binomial experiment because:
1.) Fixed number of trials (n = 3)
2.) There are only tow outcomes for each
trial (heads or tails)
3.) The outcomes are independent
4.) The probability of success (heads) is ½
in each case.
Example #1
We can solve this problem simply by
looking at the sample space:
Example #1 Cont.
In this problem:
n = 3, X = 2, p = ½ and q = ½
n!
X
n X
P(X) 
 p q
(n  X)!X!
2
3!
 1  1
p(2 heads) 
   
Example #1 Cont.
(3  2)!2!  2   2 
(32)
A survey found that one out of five
Americans say he or she has visited a
doctor in any given month. If 10 people
are selected at random, find the
probability that exactly 3 will have visited
a doctor last month.
n=
X=
Example #2
p=
q=
A survey from Teenage Research Unlimited
(Northbrook, Illinois) found that 30% of
teenage consumers receive their spending
money from part-time jobs. If 5 teenagers
are selected at random, find the probability
that at least 3 of them have part time jobs.
(To find the probability of at least, we must
find the probability of 3, 4, and 5 and then
add them together)
Example #3
Public Opinion reported that 5% of
Americans are afraid of being alone in a
house at night. If a random sample of 20
Americans is selected, find the probability
there are exactly 5 people in the sample
who are afraid of being alone at night.
Example #4
Twenty-five percent of the customers
entering a grocery store between 5 PM
and PM use an express checkout.
Consider 5 randomly selected customers
and let x denote the number among the
five who use the express checkout.
What is p(2)?
What is (P<=1)
What is P(>=2)
Example #5
Mean:
  n p
Variance:
  n pq
2
Standard Deviation:
 
2
Mean, Variance, and Standard
Deviation for the Binomial Distribution
A coin is tossed 4 times. Find the mean,
variance, and standard deviation.
A die is rolled 480 times. Find the mean,
variance and standard deviation.
Example #6 and #7
Geometric Distribution
Chapter 7 Section 5
A geometric random variable is defined as
the number of trials (x) until the first
success is observed. The probability
distribution of x is called the geometric
probability distribution.
Geometric Random Variable
p(x)  q
x1
p
Geometric Probability Distribution
Equation
P(x  n)  1 q
n
And
P(x  n)  1 (1 q )
n
Cumulative Geometric Distribution
Equation
Expected value
1
E(x) 
p

You have left your lights on and your car
battery has died. Assume that 20% of
students who drive to school carry jumper
cables. What is the probability that the first
student you ask has jumper cables?

What is the probability that 3 or fewer
students must be stopped?

What is the expected number before
someone will have jumper cables?
Example #1
One out of seven scratch off lottery tickets has
a prize. You keep buying tickets until you
win a prize, then you stop.
a.
b.
c.
d.
Find the probability that you buy 4 tickets.
Find the probability that you buy 3 or fewer
tickets
Find the probability that you buy more than
four tickets
What is the expected number of tickets
bought?
Example #2