CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 4. Discrete Probability Distribution

Standard Deviation:σ=

√

n pq=

√

n p( 1 −p)

Example:

A poll of twenty voters is taken to determine the number in favor of a certain candidate for mayor. Suppose that 60%
of all the city’s voters favor this candidate.

a. Find the mean and the standard deviation ofx.

b. Find the probability thatx≤10.

c. Find the probability thatx>12.

d. Find the probability thatx=11.

Solution:

a. Since a sample of twenty was randomly selected, it is likely thatxis a binomial random variable. Of course,x
here would be the number of the twenty who favor the candidate. The probability is 60%= 0 .6, the fraction of the
total voters who favor the candidate. Therefore, to calculate the mean and the standard deviation,

μ=n p= 20 (. 6 ) = 12

σ^2 =n p( 1 −p) = 20 (. 6 )(. 4 ) = 4. 8

The standard deviation

σ=

√

4. 8 = 2. 2

b. To calculate the probability forp(x),

p(x≤ 10 ) =p( 0 )+p( 1 )+p( 2 )+...+p( 10 )

or

p(x≤ 10 ) =

10

∑

x= 0

p(x) =

10

∑

x= 0

(

20

x

)

(. 6 )x(. 4 )^20 −x

As you can see, this can be very tedious calculations and it is best to resort tables or calculators. If you are using a
table, look up theCumulative Binomial Probability Table. To findp(x≤ 10 )forn=20 andp= 0 .6, we first find
the column that corresponds top= 0 .6 and then the row corresponding fork=10. The value is

p(x≤ 10 ) = 0. 245

However, please see the box below (Technology Note) to learn more about other options.

c. To find the probability thatp>12, the formula says,

p(x> 12 ) =p( 13 )+p( 14 )+...+p( 20 ) =

20

∑

x= 13

p(x)