Normal Approximation: Binomial

01Normal Approximation: Binomial

Approximation Conditions

Decide when a normal approximation is suitable and recognise its limits.

Replace a discrete shape with a smooth curve

A binomial distribution can be approximated by a normal distribution when the number of trials $n$ is large and the success probability $p$ is sufficiently close to $0.5$ . The binomial shape is then reasonably symmetric and bell-shaped.

Both conditions matter. A small number of trials leaves a coarse set of possible counts, while $p$ near $0$ or $1$ can produce a strongly skewed distribution that a symmetric curve represents poorly.

Suitable candidate

$X\sim\mathrm B(200,0.4)$ has many trials and a success probability reasonably close to $0.5$ .

Questionable candidate

$W\sim\mathrm B(8,0.05)$ has few trials and a success probability near zero. A normal approximation is not appropriate.

A normal model makes interval probabilities and inverse probability calculations convenient, but the answer remains an approximation.

02Normal Approximation: Binomial

Normal Parameters

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Preserve the centre and spread

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Y\sim\mathrm N\bigl(np,\,np(1-p)\bigr).

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03Normal Approximation: Binomial

Continuity Correction

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Give each integer a half-unit on either side

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The count $7$

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Binomial event	Normal event
$X=k$	$k-0.5\lt Y\lt k+0.5$
$X\leq k$	$Y\lt k+0.5$
$X\lt k$	$Y\lt k-0.5$
$X\geq k$	$Y\gt k-0.5$
$X\gt k$	$Y\gt k+0.5$

04Normal Approximation: Binomial

Approximating Probabilities

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Model, correct, calculate

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