IXL - Use normal distributions to approximate binomial distributions (Precalculus practice)

Data scientists studying online safety recently claimed that

33%

of users over age 40 have an email password containing the user's actual name. Assuming these findings are accurate, if 69 email passwords of users over age 40 are selected at random, what is the probability that exactly 20 of the passwords contain the users' actual names?

Use a normal distribution to approximate the answer and write it as a decimal rounded to the nearest hundredth.

ref_doc_title.

Learn with an example

question

Caden hosts a trivia night at his house every Thursday. He generates questions using a trivia website. The website claims that there is a

33%

chance that a generated question is a history question. If the claim is correct, and Caden randomly generates 97 questions during the first round of trivia, what is the probability that exactly 43 will be history questions?

Use a normal distribution to approximate the answer and write it as a decimal rounded to the nearest hundredth.

key idea

A binomial experiment consists of identical and independent trials. In each trial there are two outcomes: success and failure.

In a binomial experiment, the number of successes is a random variable. Its probability distribution is called the binomial

distribution.

The binomial distribution depends on

the number of trials, and

the probability of success. Sometimes

–

the probability of failure, is written as

If a random variable has a binomial distribution, its mean, variance, and standard deviation are given by:

mean:

𝜇

variance:

𝜎

standard deviation:

𝜎

A binomial distribution can be approximated by a normal distribution if:

n
≥
30
n
p
≥
5
n
q
≥
5

If these conditions hold, and

is a normally distributed random variable with the same mean and standard deviation as a binomial variable

then:

P
(
X
=
a
)
≈
P
(
a
–
0.5
<
Y
<
a
+
0.5
)
P
(
a
<
X
<
b
)
≈
P
(
a
+
0.5
<
Y
<
b
–
0.5
)
P
(
a
≤
X
≤
b
)
≈
P
(
a
–
0.5
<
Y
<
b
+
0.5
)

The 0.5 terms added to or subtracted from

and

are called continuity correction terms.

solution

Use the random variable whose value is the number of history questions. Call it

start of large bold text, Interpret the situation as a binomial experiment., end of large bold text

Identify the trials.

A trial is generating a trivia question.

Identify a success.

A trial is a success if the question is a history question. It is a failure if the question is not a history question.

The question describes a binomial experiment in which the number of trials is 97, the probability of success is 0.33, and the probability of failure is

–

0.33

0.67

So,

0.33

and

0.67

start of large bold text, Find the probability., end of large bold text

You want to find the probability that

is exactly 43:

(

)

The following conditions hold, so the distribution of

can be approximated by a normal distribution:

n
≥
30
(since
n
=
97
)
n
p
≥
5
(since
n
p
=
97
0.33
=
32.01
)
n
q
≥
5
(since
n
q
=
97
0.67
=
64.99
)

Find the mean and standard deviation of

𝜇

0.33

32.01

𝜎

0.33

0.67

4.631058…

Approximate the probability using a normally distributed random variable.

Define

to be a normally distributed random variable with mean

𝜇

32.01

and standard deviation

𝜎

4.631058…

The distribution of

approximates the distribution of

if continuity correction terms are included:

(

)

≈

(

–

0.5

)

(

)

≈

(

–

0.5

)

(

)

≈

(

42.5

43.5

)

Find the

-values,

and

To find

(

42.5

43.5

)

use the standard normal variable

–

𝜇

𝜎

Calculate the

-values,

and

for

42.5

and

43.5

Use the definition of

along with

𝜇

32.01

and

𝜎

4.631058…

42.5

–

𝜇

𝜎

42.5

–

32.01

4.631058…

2.265141…

43.5

–

𝜇

𝜎

43.5

–

32.01

4.631058…

2.481074…

The two

-values

are

2.265141…

and

2.481074…

Find the probability that

is between

and

Since

is a standard normal variable, use a calculator to find the probability that it is between

2.265141…

and

2.481074…

(

2.265141…

2.481074…

)

≈

0.01

So,

(

)

≈

0.01

Looking for more? Try this:

$Confetti$

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Work it out

Not feeling ready yet? These can help:

EE.6Find values of normal variables

EE.2Mean, variance, and standard deviation of binomial distributions

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