Download 7.5 The Normal Curve Approximation to the Binomial Distribution

Advanced Math Topics
7.5 The Normal Curve Approximation
to the Binomial Distribution
Find the probability of getting at least 8 heads on 10 flips of a coin.
From section 6.5, The Binomial Distribution Formula, we would solve like this…
10C8(.50)
8(.50)2
9
1
10C9(.50) (.50)
10
0
10C10(.50) (.50)
= .0439
= .0098
= .0010
Add these probabilities up to get 0.0547.
The probability of getting 8 or more heads is 5.47%.
This took some calculation. We can use the normal curve to help us answer the
same question. This is especially helpful when the number of trials becomes large.
Find the probability of getting at least 8 heads on 10 flips of a coin.
Let’s take a look at the probability distribution:
x
p(x)
0
.0010
It is a normal distribution.
1
.0098
The question asks for
8 or more heads to be
flipped .
2
.0439
3
.1172
4
.2051
5
.2461
6
.2051
7
.1172
8
.0439
9
.0098
10
.0010
The histogram of the probability distribution is shown here:
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Thus, we must find the area under the curve to the right of 7.5.
Hint: When solving these problems, x is 0.5 more or less toward
the outside of the interval you are solving.
Find the mean and standard deviation using the formulas in section 6.6.
μ = np
μ = 10(.5)
μ=5
σ =√npq
x - μ 7.5 – 5
σ =√10(.5)(.5) z = σ = 1.5811 = 1.58
σ =1.5811 Find the probability from the mean to a z of 1.58 using the table in the appendix.
.5000 - .4429 = .0571
The probability of getting 8 or more heads is 5. 71%.
A TV company telephones 5,000 people and will cancel
a show if less than 1900 viewers are watching the show.
Find the probability that the show will be canceled if 40%
of viewers watch the show.
Steps:
1) Find the mean and standard deviation.
μ = np
σ = √npq
μ = 5000(.40)
μ = 2000
σ = √5000(.40)(.60)
σ = 34.6410
2) Find out your interval.
We are interested in the probability of less
than 1900 viewers. (0,1,2….1899 viewers)
3) Add OR subtract 0.5 to the OUTSIDE
of your interval.
To the outside of the interval is +0.5.
Thus, x = 1899.5.
4) Find your z value. z = x - μ
σ
5) Locate the probability in the Standard
Normal curve table.
z = x - μ = 1899.5 – 2000 = -2.90
σ
34.6410
The probability from the mean to 1899.5 is
0.4981.
6) Find the correct probability for the interval. .5000 – 0.4981 = 0.0019
It may help to draw a bell curve.
The probability that the show will be canceled
is 0.19%.
From the HW P. 385
2) If 68% of University students have type O blood, what is the probability that a random
sample of 700 students will contain 510 or more students with type O blood?
.34%
From the HW P. 385
10) 8% of people who borrow from the library do no return the books on time. If 450
people borrowed books from the library today, find the probability that 28 of them
will not return the books on time.
2.74%
HW
P. 385 #1-12