Paper 6. A binomial with a huge \(n\) and a tiny \(p\) behaves like a Poisson, and a Poisson with a large mean behaves like a normal. Knowing when each swap is allowed, and how to correct for continuity, turns long calculations into short ones.
If \(X \sim B(n, p)\) with \(n > 50\) and \(np < 5\), use \(X \approx \text{Po}(np)\). If \(X \sim \text{Po}(\lambda)\) with \(\lambda > 15\), use \(X \approx N(\lambda, \lambda)\): the variance equals the mean, and a continuity correction of \(0.5\) is needed because a discrete count is replaced by a continuous variable.
Fig. 1: Po(20) (bars, drawn to scale) and the normal curve N(20, 20). The teal bars are X ≤ 15; the approximating area stops at 15.5, half-way through the next bar
Section 01
Binomial to Poisson.
When \(n\) is large and \(p\) is small, successes are rare events scattered among many trials, which is exactly the Poisson picture. The mean \(np\) carries over and the variance \(np(1-p)\) is almost \(np\) because \(1 - p \approx 1\).
Exact
Conditions
Approximation
\(B(n, p)\)
\(n > 50\) and \(np < 5\)
\(\text{Po}(np)\)
\(B(n, p)\), \(p\) close to 1
\(n > 50\) and \(n(1-p) < 5\)
count failures: \(\text{Po}(n(1-p))\)
Worked example A rare defect
1.5% of bolts made by a machine are faulty. A random sample of 200 bolts is taken. Use a suitable approximation to find the probability that at most 2 are faulty.
\(X \sim B(200, 0.015)\). Here \(n = 200 > 50\) and \(np = 3 < 5\), so use \(X \approx \text{Po}(3)\).
\(= 0.4232\). (The exact binomial value is \(0.4215\), so the approximation is good.)
Answer: \(0.423\) (3 s.f.)
Worked example When \(p\) is close to 1
97% of eggs in a large batch are undamaged. A box of 80 eggs is chosen at random. Find, using an approximation, the probability that at least 78 eggs are undamaged.
Undamaged eggs: \(B(80, 0.97)\), with mean 77.6. Neither approximation fits this directly.
State the conditions with numbers: "\(n = 200 > 50\) and \(np = 3 < 5\)". Writing "\(n\) is large and \(p\) is small" with no values often loses the mark, and using \(\text{Po}(np(1-p))\) is wrong: the Poisson parameter is the mean, \(np\).
Section 02
Poisson to normal.
As \(\lambda\) grows, the Poisson bars become symmetric and bell-shaped (Fig. 1). For \(\lambda > 15\), use \(Y \sim N(\lambda, \lambda)\), remembering that the variance is \(\lambda\), so the standard deviation is \(\sqrt{\lambda}\).
Poisson event
Includes
Normal event
\(X \le 15\)
15
\(Y < 15.5\)
\(X < 15\)
up to 14
\(Y < 14.5\)
\(X \ge 25\)
25 upwards
\(Y > 24.5\)
\(X > 25\)
26 upwards
\(Y > 25.5\)
Worked example A lower tail
\(X \sim \text{Po}(20)\). Use a suitable approximation to find \(\text{P}(X \le 15)\).
\(\lambda = 20 > 15\), so \(X \approx Y \sim N(20, 20)\).
\(X \sim \text{Po}(22)\). Use a normal approximation to find \(\text{P}(X < 19)\).
\(X < 19\) means \(X \le 18\), so \(\text{P}(Y < 18.5)\) with \(Y \sim N(22, 22)\). \(z = \dfrac{18.5 - 22}{\sqrt{22}} = -0.746\), giving \(1 - 0.7722 = 0.228\).
Examiner trap
The standard deviation of \(N(20, 20)\) is \(\sqrt{20}\), not 20. Dividing by 20 is the commonest error in this topic, closely followed by forgetting the continuity correction altogether.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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Which binomial distribution is best approximated by a Poisson distribution?
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\(X \sim \text{Po}(25)\). Which normal distribution approximates \(X\)?
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\(X \sim \text{Po}(40)\) is approximated by \(Y \sim N(40, 40)\). Which event replaces \(X < 30\)?
Quick check+10 XP
Why is no continuity correction used when \(B(n, p)\) is approximated by \(\text{Po}(np)\)?
Section 03
Choosing the right approximation.
Check the parameters before choosing. Paper 6 questions often say "use a suitable approximation" and award a mark just for justifying the choice.
\(B(n, p)\), \(n > 50\), \(np < 5\): Poisson with \(\lambda = np\). No continuity correction (Poisson is discrete too).
\(B(n, p)\), \(np > 5\) and \(nq > 5\): normal \(N(np, npq)\) with continuity correction (from Paper 5).
\(\text{Po}(\lambda)\), \(\lambda > 15\): normal \(N(\lambda, \lambda)\) with continuity correction.
\(\text{Po}(\lambda)\), \(\lambda\) small: no approximation; use the Poisson formula directly.
Examiner trap
Do not apply a continuity correction when approximating binomial by Poisson. Both are discrete, so \(\text{P}(X \le 2)\) stays \(\text{P}(X \le 2)\).
Stage 2 · Exam
Exam-style questions
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\(X \sim B(120, 0.02)\). Which distribution is the most suitable approximation to \(X\)?
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\(X \sim \text{Po}(18)\). Using a normal approximation, find \(\text{P}(X \ge 20)\).
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\(X \sim B(250, 0.014)\). Use a suitable approximation to find \(\text{P}(X > 4)\).
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98% of seeds of a certain type germinate. 150 seeds are sown. Using a suitable approximation, find the probability that at most 146 seeds germinate.
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