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.
Understanding Statistics: Binomial vs Poisson Approximation
The reason the approximation works comes from the shape of repeated rare trials. A binomial count has a fixed maximum of n, since no more than n successes can occur. When p is tiny, counts near that upper limit are extraordinarily unlikely.
Most of the useful probability sits near zero and near the average count. In that lower part of the distribution, the detailed limit of n trials matters very little.
The factor for failures, one minus p raised to a large power, settles into the exponential factor used by the Poisson model. This is a limiting result, not a claim that the two distributions are identical.
Matching the mean is necessary, but it does not guarantee a good approximation. The important difference is spread. A binomial distribution has variance equal to n times p times one minus p.
A Poisson distribution has variance equal to lambda. Since lambda equals n times p, the Poisson variance is slightly larger. When p is very small, one minus p is close to one, so the difference is small.
If p is not small, the Poisson model spreads probability too widely. It can then overestimate the chance of counts far from the mean and give noticeably inaccurate answers.
Consider checking a large batch of products where each product has a one in five hundred chance of a particular defect. If ten thousand products are checked, the expected number of defects is twenty. A Poisson model with mean twenty can estimate the chance of finding exactly fifteen defects or at most twenty defects.
This is useful in quality control because the calculation depends on the expected count rather than on a huge number of possible product selections. The same reasoning appears in insurance claims, DNA mutations, website errors, and particles detected by scientific instruments. The setting may look different, but the event being counted must remain rare for each opportunity.
Students should check the assumptions before selecting the approximation. There must be a clear number of opportunities, and the success probability should be roughly the same for each one. Independence matters because events that cluster do not behave like a simple Poisson process.
For example, a machine fault can make several defective items in a row. Disease cases can rise together when people share exposure.
These patterns often produce more variation than the Poisson model expects. A changing event rate creates a similar problem, such as calls arriving much faster during one part of the day.
It is worth comparing answers when calculation tools are available. Find the exact binomial probability, then find the Poisson estimate using the same expected count. Look at the absolute difference and at whether the estimate is too high or too low.
Pay special attention to tail probabilities, such as at least a certain number, because small errors in individual probabilities can build up when many values are added. Remember that a Poisson approximation can assign a tiny probability to counts above n, even though a binomial count cannot exceed n. That is harmless only when such impossible counts are practically zero.
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.