4.4. The Binomial Probability Distribution http://www.ck12.org
b. The probability at most one customer means
p(x≤ 1 ) =p( 0 )+p( 1 )=(
5
0
)
(. 2 )^0 (. 8 )^5 −^0 +
(
5
1
)
(. 2 )^1 ( 08 )^5 −^1
= 0. 328 + 0. 410 =. 738
c. The probability of at least one sale is
p(x≥ 1 ) =p( 1 )+p( 2 )+p( 3 )+p( 4 )+p( 5 )We can now apply the binomial probability formula to calculate the five probabilities. However, we can save time
by calculating the complement of the probability,
p(x≥ 1 ) = 1 −p(x< 1 ) = 1 −p(x= 0 )1 −p( 0 ) = 1 −(
5
0
)
(. 2 )^0 (. 8 )^5 −^0
= 1 − 0. 328 = 0. 672
This tells us that the salesperson has a chance of 67.2% of making at least one sale in five attempts.
d. Here, we are asked to determine the probability distribution for the number of salesxin five attempts. So we need
to computep(x)forx= 1 , 2 , 3 , 4 ,and 5. We use the binomial probability formula for each value ofx. The table
below shows the probabilities.
x p(x)
0 0. 328
1 0. 410
2 0. 205
3 0. 051
4 0. 006
5 0. 00032Figure:The probability distribution for the number of sales.
In many applications of the binomial distribution, it is necessary that we know how to calculate themeanand the
standard deviation.To compute these we use the following formulas shown in the box below.
Mean, Variance, and Standard Deviation for a Binomial Random Variable
Mean:μ=n p
Variance:σ^2 =n pq=n p( 1 −p)