What is wrong direction of the correction?

For X ≥ k shift down to k - 0.5; for X ≤ k shift up to k + 0.5. Sketch to check.

What is wrong variance for the Poisson normal approximation?

Both the mean and variance equal λ.

Apply the continuity correction to P(X ≥ 20) for a normal approximation. [1 mark]

SingaporeMathsSyllabus dot point

When can the binomial or Poisson be approximated by the normal or Poisson, and how is the continuity correction applied?

Approximate the binomial by the Poisson or the normal, and the Poisson by the normal, under stated conditions, applying a continuity correction where appropriate

A focused answer to the H2 Mathematics outcome on distribution approximations. The conditions for the Poisson and normal approximations to the binomial and the normal approximation to the Poisson, and the continuity correction.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

SEAB wants you to approximate one distribution by another under the stated conditions: the binomial by the Poisson (large $n$ , small $p$ ), the binomial by the normal (large $n$ , $p$ not too extreme), and the Poisson by the normal (large $\lambda$ ), applying a continuity correction when approximating a discrete variable by a continuous one.

The answer

Poisson approximation to the binomial

When $n$ is large and $p$ is small (a common guide is $n > 50$ and $p < 0.1$ ), with $\lambda = np$ moderate,

\mathrm{B}(n, p) \approx \mathrm{Po}(np).

No continuity correction is needed because both are discrete.

Normal approximation to the binomial

When $n$ is large with $np > 5$ and $n(1 - p) > 5$ ,

\mathrm{B}(n, p) \approx \mathrm{N}\big(np,\ np(1 - p)\big).

A continuity correction is needed because a discrete variable is being approximated by a continuous one.

Normal approximation to the Poisson

When $\lambda$ is large (a common guide is $\lambda > 10$ ),

\mathrm{Po}(\lambda) \approx \mathrm{N}(\lambda, \lambda).

Again a continuity correction applies.

The continuity correction

To approximate a discrete count by a continuous normal, widen each integer by half a unit:

$\mathrm{P}(X \geq k) \approx \mathrm{P}(X > k - 0.5)$
$\mathrm{P}(X \leq k) \approx \mathrm{P}(X < k + 0.5)$
$\mathrm{P}(X = k) \approx \mathrm{P}(k - 0.5 < X < k + 0.5)$

Forgetting the half-unit shift is the commonest error.

Worked example

A factory finds $20\%$ of items need rework. In a batch of $80$ , use a suitable approximation to find the probability that fewer than $12$ need rework.

Step 1: Choose the approximation

$X \sim \mathrm{B}(80, 0.2)$ . Check $np = 16 > 5$ and $n(1-p) = 64 > 5$ , so use the normal: $X \approx \mathrm{N}(16, 12.8)$ (variance $np(1-p) = 80 \times 0.2 \times 0.8 = 12.8$ ).

Step 2: Apply the continuity correction

"Fewer than $12$ " means $X \leq 11$ , so $\mathrm{P}(X < 11.5)$ after correction.

Step 3: Standardise

z = \frac{11.5 - 16}{\sqrt{12.8}} = \frac{-4.5}{3.578} \approx -1.258

Step 4: Read the probability

\mathrm{P}(Z < -1.258) = 1 - \mathrm{P}(Z < 1.258) \approx 1 - 0.8957 = 0.104

Examples in context

Example 1. Polling. A national poll of thousands of voters models the count supporting a party with $\mathrm{B}(n, p)$ but computes it via the normal approximation, since $n$ is huge and $p$ moderate, with a continuity correction for exact counts.

Example 2. Call centre load. Calls at $\lambda = 50$ per hour are well approximated by $\mathrm{N}(50, 50)$ , letting a manager estimate the probability of exceeding capacity using normal tables rather than summing many Poisson terms.

Try this

Q1. State the conditions for approximating the binomial by the Poisson. [2 marks]

Cue. $n$ large and $p$ small (so $np$ is moderate), with $\lambda = np$ .

Q2. Apply the continuity correction to $\mathrm{P}(X \geq 20)$ for a normal approximation. [1 mark]

Cue. $\mathrm{P}(X > 19.5)$ .

Q3. For $\mathrm{B}(200, 0.5)$ , state the normal approximation's mean and variance. [2 marks]

Cue. Mean $100$ , variance $200 \times 0.5 \times 0.5 = 50$ .

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original5 marksA biased coin lands heads with probability

0.4

. It is tossed

100

times. Using a suitable approximation, find the probability of at least

50

heads.

Show worked answer →

$X \sim \mathrm{B}(100, 0.4)$ . Since $n$ is large and $np = 40 > 5$ , $n(1-p) = 60 > 5$ , use the normal approximation: $X \approx \mathrm{N}(40, 24)$ (mean $np = 40$ , variance $np(1-p) = 24$ ).

"At least $50$ " with continuity correction: $\mathrm{P}(X \geq 50) \approx \mathrm{P}(X > 49.5)$ .

Standardise: $z = \dfrac{49.5 - 40}{\sqrt{24}} = \dfrac{9.5}{4.899} \approx 1.939$ .

$\mathrm{P}(Z > 1.939) \approx 1 - 0.9737 = 0.0263$ .

Markers reward the normal approximation with correct parameters, the continuity correction to $49.5$ , standardising, and the tail probability.

Original4 marksA rare disease affects

0.5\%

of a population. In a sample of

400

, use a suitable approximation to find the probability of exactly

1

case.