AP Statistics: Section 8.1A
Binomial Probability
There are four conditions to a binomial setting:
1. Each observation falls into one of just two
categories: _________
success or ________.
failure
2. There is a ______
finite number of observations, __.
n
3. These n observations are all ____________.
independen t
4. The probability of success, __,
p is _________
constant
for each observation.
If data are produced in a binomial setting,
then the random variable X = the number
of successes is called a binomial random
variable and the probability distribution of
X is called a binomial distribution which is
abbreviated ______.
B (n, p )
Example 1: Determine if each of
the following situations is a
binomial setting. If so, state the
probability distribution for X. If
not, state which of the 4 conditions
above is not met.
Situation 1: Blood type is inherited. If both parents
carry genes for the O and A blood types, each child has
probability 0.25 of getting two O genes and so having
blood type O. A couple’s 5 different children inherit
independently of each other. Let X = number of
children with type O blood.
B ( 5 ,. 25 )
Situation 2: Deal 10 cards from a shuffled
deck and let X = the number of red cards.
No, probabilit ies are not constant.
26 26
25
,
or

52 51
51
Situation 3: An engineer chooses a SRS of 10 switches
from a shipment of 10,000 switches. Suppose that
(unknown to the engineer) 10% of the switches in the
shipment are bad. Let X = the number of bad
switches in the sample.
B (10 ,. 1)
****When choosing an SRS from a
population that is much larger than
the sample, the observations are
considered independent.
Binomial Probability: If X has a binomial distribution
with n observations and probability p of success on
each observation, the possible values of X are 0, 1, 2, 3,
. . . , n. If k is any one of these values, then
n k
nk
P ( x  k )    ( p )(1  p )
k 
n
 
k 
The notation
is read n choose k
and means the number of possible
ways to choose k objects from a group
of n objects.
Ck
It is also written _____
n
n
  
k 
n!
k ! ( n  k )!
TI 83 / 84 : MATH
PRB 3 : n C r
The notation n! is read n factorial and means n(n - 1)(n - 2)    (3)(2)(1)
So 6!  6  5  4  3  2  1  720
By definition
0!  1
TI 83 / 84 : MATH
PRB
4 :!
Example 2 (combinations): How many
different ways can we choose a
subcommittee of size 3 from a student
council that has 7 members?
7
7!
210
  

 35
6
 3  3!4!
7 n C r 3  35
Example 3: You randomly guess the answers
to10 multiple choice questions which have 5
possible answers. What is the probability of
getting exactly 6 correct answers?
B (10 ,. 2 )
with x  6
 10  6
4
  (. 2 )(. 8 ) 
6 
. 0055
Binomial Probability on the
TI83/84:
2
nd
VARS
DISTR 0 : binomialpd f
ENTER
binomialpd f ( n , p , x )
Example 4: Consider situation 3 in example 1.
Find the probability that in an SRS of size 10, no
more than 1 switch fails.
B (10 ,. 1) where x  0 or 1
binomialpd f (10 ,. 1, 0 )  . 3487
binomialpd f (10 ,. 1,1)  . 3874
.7361
Example 5: Corinne is a basketball player who makes 75% of
her free throws over the course of a season. In a big game,
Corinne shoots 12 free throws and makes only 5 of them. Is it
unusual for Corinne to perform this poorly?
Note: We actually want the probability of making a basket on
at most 5 free throws.
B (12 ,. 75 ) where x  0,1,2,3,4 or 5
Cumulative Binomial Probability on the TI83/84:
2
nd
VARS
DISTR
A : binomialcd f
binomialcd f ( n , p , largest x value)
binmialcdf
(12 ,. 75 , 5 )  . 0143
Any difference in answers between
the manual calculation and using
your calculator is due to rounding
error.