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Normal Approximation to
the Binomial Distribution
To understand and know how to calculate
probabilities when asked to use a Normal
approximation to the Binomial Distribution.
To know the conditions under which this
approximation can + should be used.
Continuity Correction
Because the binomial distribution is discrete
and the normal distribution is continuous we
need to apply a continuity correction. (last
Conditions of Use
If X~Bi(n,p)
then we can use the normal distribution if
n is large (>20)
p is around 0.5
This should lead to np>5 and nq>5
Then X~N(np,npq)
The larger n is, the better approximation we
What’s the point in approximating?
Calculations may be less tedious.
Calculations will be made easier and quicker.
Imagine, X~Bi(1000,0.45) find P(X≤342)
Tables are not big enough.
Lots of calculations.
Example 1
Find the probability of 4,5,6 or 7 heads when
a dice is rolled 12 times, using:
(i) the binomial distribution.
(ii) the normal distribution.
Example 2: Exam Question
Drug manufacturer claims that a certain drug
cures on average 80% of the time. He
accepts the claim if k or more patients are
(i) State the distribution of X
(ii) Find the probability that the claim will be
accepted when 15 individuals are tested and
k is set at 10.
(iii) A more extensive trial is undertaken on
100 patients. The distribution may now be
approximated by the Normal distribution.
State the approximating distribution.
(iv) Using the approximating distribution,
estimate the probability that the claim will be
rejected if k is set at 75.
Work: Ex 2B pg 54