Page 311 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 15.5 Normal Approximation to Binominal Distribution

Normal distribution can be used as an approximation to binomial distribution when n is large
and neither p nor q is very small. If X denotes the number of successes with probability p of a
success in each of the n trials, then X will be distributed approximately normally with mean np
and standard deviation npq .

X np
Further, z ~ N 0,1 .

npq
It may be noted here that as X varies from 0 to n, the standard normal variate z would vary
from to because
lim np lim np
when X = 0,
n npq n q

lim n np lim nq lim nq
and when X = n,
n npq n npq n p

Correction for Continuity

Since the number of successes is a discrete variable, to use normal approximation, we have make
corrections for continuity. For example,

1 1
P(X X X ) is to be corrected as P X 1 X X 2 , while using normal approximation
1 2 2 2
to binomial since the gap between successive values of a binomial variate is unity. Similarly,
1 1
P(X < X < X ) is to be corrected as P X 1 X X 2 , since X < X does not include X and
1 2 2 2 1 1
X < X does not include X .
2 2
Note: The normal approximation to binomial probability mass function is good when n 50 and neither
p nor q is less than 0.1.

Example: An unbiased die is tossed 600 times. Use normal approximation to binomial to
find the probability obtaining
1. more than 125 aces,
2. number of aces between 80 and 110,
3. exactly 150 aces.
Solution:
Let X denote the number of successes, i.e., the number of aces.

1 1 5
np 600 100 and npq 600 9.1
6 6 6
1. To make correction for continuity, we can write

P(X > 125) = P(X > 125 + 0.5)

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