A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. A Poisson distribution models the number of rare events in a fixed interval when events occur independently at a constant average rate. The Poisson approximation is useful because binomial calculations can be time consuming when n is large and p is small.
It lets you replace many-trial counting problems with a simpler formula using only λ = np.
The approximation works best when the expected number of successes stays moderate while each individual success is rare. In this situation, the binomial bars become very close to the Poisson probabilities with the same mean λ. A common rule of thumb is to use the Poisson approximation when n ≥ 20 and p ≤ 0.05, with better accuracy when n is larger and p is smaller.
This method is widely used for defects, misprints, rare diseases, calls arriving, and other low-probability event counts.
Key Facts
- Binomial probability: P(X = k) = C(n, k)p^k(1 - p)^(n - k)
- Poisson probability: P(X = k) = e^(-λ)λ^k / k!
- Use λ = np to match the Poisson mean to the binomial mean.
- Binomial mean and variance: μ = np and σ^2 = np(1 - p)
- Poisson mean and variance: μ = λ and σ^2 = λ
- Poisson approximation is best when n is large, p is small, and λ = np is not too large.
Vocabulary
- Binomial distribution
- A probability distribution for the number of successes in n independent trials with the same success probability p.
- Poisson distribution
- A probability distribution for the number of rare independent events occurring in a fixed interval with average rate λ.
- Lambda
- The symbol λ represents the average number of events and equals np when approximating a binomial distribution.
- Rare event
- A rare event is an outcome with a small probability on each individual trial.
- Approximation error
- Approximation error is the difference between the exact binomial probability and the estimated Poisson probability.
Common Mistakes to Avoid
- Using λ = n + p instead of λ = np is wrong because λ must represent the expected number of successes.
- Applying the Poisson approximation when p is not small is wrong because the binomial shape may no longer match the Poisson shape closely.
- Forgetting that trials must be independent is wrong because dependence changes the probability pattern and can make both binomial and Poisson models inappropriate.
- Comparing only the means and ignoring variance is wrong because the binomial variance is np(1 - p), while the Poisson variance is λ.
Practice Questions
- 1 A factory makes 200 light bulbs, and each bulb has a 0.01 probability of being defective. Use the Poisson approximation to estimate the probability that exactly 3 bulbs are defective.
- 2 A website has 500 visitors, and each visitor has a 0.004 probability of making a purchase. Find λ and use the Poisson approximation to estimate the probability of at least 1 purchase.
- 3 A quiz has 10 questions, and a student guesses each answer with probability 0.25 of being correct. Explain whether a Poisson approximation to the binomial distribution is appropriate.