Normal Approximation: Binomial

For a binomial distribution B(n, p),
if n is big,
then the data looks like
a normal distribution N(np, npq).

Using this property
is the normal approximation
to the binomial distribution.

So, when using the normal approximation
to a binomial distribution,

First change B(n, p) to N(np, npq).

Then standardize N(np, npq) to N(0, 1²).

np: Expected value, mean
npq: (Standard Deviation)²

First, write the given condition as
B(n, p).

Getting a head of a coin
isn't affected by the previous trial.
So this is an independent event.

This is repeated.
So this is a binomial experiment.

n = 400

So write
B(400.

The probability of getting a head of a coin is
1/2.

So p = 1/2.

So write
1/2).

So the given binomial experiment is
B(400, 1/2).

B(400, 1/2)
p = 1/2

Then q = 1 - 1/2 = 1/2.

B(400, 1/2)
n = 400
p = 1/2

q = 1/2

Then, by the normal approximation,
this becomes
N(400⋅[1/2], 400⋅[1/2]⋅[1/2]).

400⋅[1/2] = 200

400⋅[1/2]⋅[1/2] = 100

100 = 10²

So N(200, 10²).

The mean (expected value) is 200.
The standard deviation is 10.

The probability of
getting a head 185 ~ 210 times
is P(185 ≤ X ≤ 210).

Find the z-score of 185.

N(200, 10²)
x = 200
σ = 10

Then Z = [185 - 200]/10.

185 - 200 = -15

-15/10 = -1.5

So the z-score of X = 185 is
Z = -1.5.

Find the z-score of 210.

N(200, 10²)
x = 200
σ = 10

Then Z = [210 - 200]/10.

210 - 200 = 10

10/10 = 1

So the z-score of X = 210 is
Z = 1.

X = 185 is Z = -1.5.
X = 210 is Z = 1.

So
P(185 ≤ X ≤ 200) = P(-1.5 ≤ Z ≤ 1).

Draw the normal distribution curve
like this.

Color the region
under the curve -1.5 ≤ Z ≤ 1.

The blue colored area is
P(-1.5 ≤ Z ≤ 0).

The left side and the right side are the same.
So P(-1.5 ≤ Z ≤ 0) = P(0 ≤ Z ≤ 1.5).

See the given z-score table.
P(0 ≤ Z ≤ 1.5) = 0.4332

So the blue colored area is
0.4332.

The green colored area is
P(0 ≤ Z ≤ 1).

See the given z-score table.
P(0 ≤ Z ≤ 1) = 0.3413

So the green colored area is
0.3413.

P(-1.5 ≤ Z ≤ 1) is the colored area
under the curve.

So
P(-1.5 ≤ Z ≤ 1)
= 0.4332 + 0.3413
= 0.7745.

So 0.7745, 77.45%, is the answer.