6.3 Notes - Binomial Random Variables
Binomial Setting:
1. Each observation falls in one of two categories (Success or Failure)
 Ex. Girl or Boy
2. There is a fixed number n of observations
 Ex. n = 3
3. The n observations are all independent
 Ex. One girl has no influence on the outcome of the next
4. The probability of success, p, is the same for each observation
 Ex. P(Girl) = 0.5 P(Boy) = 0.5
The distribution of the count X of successes in the binomial setting is the binomial distribution with
parameter n and p. X is B  n, p 
B
I
N
S
Binary (2 outcomes/success or failure)
Independent
Fixed number n of trials
Same probability of success
P.D.F. – Probability Distribution Function
Finds the probability of a certain outcome
Ex. P(X = 1) or P(X = 3)
C.D.F. – Cumulative Distribution Function
Finds the probability of an interval of outcomes
Ex. P(X < 1) or P(X > 2)
Look over example on page. 390-391 titled ‘Inheriting Blood Type’
In order to calculate a Binomial probability, we first need to find the number of arrangements of k
successes in n observations (Binomial Coefficient) without actually listing all of them.
n
n!
 
 k  k ! n  k !
! = Factorial, so for example, 5!  5  4  3  2 1
6
6!
6  5  4  3  2 1 30


 15
 
 2  2! 6  2 ! 2 1 4  3  2 1 2
The number 15 represents the number of ways you can write 2 successes out of 6 trials
Binomial Probability:
If X has the binomial distribution with n observations and probability p of success, then:
n
nk
P( X  k )    p k  1  p 
k 
Mean of a Binomial Random Variable
  n p
Standard Deviation of a Binomial Random Variable
  n  p 1  p 
Example:
A dog food manufacturer is promoting a new brand of food with a rebate offer on its 25-lb. page. Each
package is supposed to contain a mail-in coupon for $4. However, the company has found that the
machine dispensing these fails to place a coupon in 10% of the bags. Enticed by the large rebate offer,
an owner of 2 dogs purchases 5 bags. Find the probability that
a. Check BINS to make sure it is Binomial
b. One of the bags will not contain a coupon.
c. At least 1 bag will fail to have a coupon
d. Find the mean and standard deviation of this binomial distribution
a.
b.
B
I
N
S
Success or Failure (Contains Rebate or Does Not Contain Rebate
Each trial is independent
Fixed n number of trials (5 bags)
Probability of Success is same (0.1)
5 1
51
P( X  1)    .1 1  .1
1 
51
 5!  1

 .1 1  .1
 1!4! 
=  5.1.9 
P( X  1)  1  P( X  0)
c.
5
0
5 0
 1    .1 1  .1
0
= 1  11.9 
4
5
= .4095
= .328
There is a 32.8% chance that one of the bags will
not contain a coupon.
There is a 40.95% chance that at least 1 bad will
fail to have a coupon.
d.
Mean   5  .1  0.5
Calculator:
S.D.   5  .11  .1  5  .1.9   .6708
2nd-Vars-0 = Binomial PDF … binompdf(n,p,k) will give you binomial probability
2nd-Vars-A = BinomialCDF … binomcdf (n,p,k) will give you cumulative probability to left
6
 2
Math-Prb-3 will give you the binomial coefficient … 6 6 nCr 2   