The normal distribution – The normal approximation to the binomial distribution

If \(X \sim B(n,p)\), we may approximate using a normal distribution when:

Then:

The standard deviation is:

\[ \sigma = \sqrt{np(1-p)} \]

The binomial distribution is discrete, but the normal distribution is continuous.

So we apply a continuity correction.

Binomial Probability	Normal Approximation
\(P(X = k)\)	\(P(k - 0.5 < X < k + 0.5)\)
\(P(X \le k)\)	\(P(X < k + 0.5)\)
\(P(X < k)\)	\(P(X < k - 0.5)\)
\(P(X \ge k)\)	\(P(X > k - 0.5)\)
\(P(X > k)\)	\(P(X > k + 0.5)\)

A biased coin has probability \(p = 0.6\) of landing heads.

The coin is tossed \(n = 100\) times.

Find \(P(X \ge 65)\) using a normal approximation.

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Step 1: Check conditions

\[ np = 100(0.6) = 60 \] \[ n(1-p) = 100(0.4) = 40 \]

Both are greater than 5, so the approximation is valid.

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Step 2: Mean and variance

\[ \mu = np = 60 \] \[ \sigma^2 = np(1-p) = 100(0.6)(0.4) = 24 \] \[ \sigma = \sqrt{24} \approx 4.90 \]

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Step 3: Continuity correction

\[ P(X \ge 65) \approx P(X > 64.5) \]

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Step 4: Standardise

\[ Z = \frac{64.5 - 60}{4.90} \] \[ Z \approx 0.92 \]

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Step 5: Use normal tables

\[ P(Z > 0.92) = 0.1788 \]

• Always check the conditions \(np \ge 5\) and \(n(1-p) \ge 5\).
• State the approximating distribution clearly.
• Always apply the continuity correction.
• Standardise using \(Z = \frac{x-\mu}{\sigma}\).
• Use correct probability notation when writing answers.