Download PP Section 8.1A

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 ____________.
independent
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, probabilities 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.
It is also written _____
n Ck
n
  
k 
n!
k!(n  k )!
TI 83 / 84 : MATH PRB 3 : n Cr
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 Cr 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
 (.2 )(.84 )  .0055
6 
Binomial Probability on the
TI83/84:
2 nd
VARS DISTR 0 : binomialpdf ENTER
binomialpdf (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
binomialpdf (10,.1,0)  .3487
binomialpdf (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 : binomialcdf
binomialcdf (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.