Counting Models
EN.553.211 Probability and Statistics for Life Sciences
Monday review and Wednesday read-ahead · September 28 and 30, 2026 · Zan Ahmad
When can we add expectations, when can we add variances, and how do the sampling assumptions change a count’s distribution? These cumulative notes connect independence and Bernoulli trials to binomial, hypergeometric, and Poisson models, with worked derivations and simulation figures.
Read or download the complete notes (PDF, 35 pages)
Class & test review slides · Detailed animated explanations
Start with a coin flip and build toward binomial, hypergeometric, and Poisson counts. The class version emphasizes the formulas and examples; the detailed version adds the derivations, Pascal’s triangle, and simulations. Switching versions takes you to the closest matching point.
Try the interactive practice quiz — 12 questions with hints and explanations
Wednesday starts at cereal example, part (b), page 16. Resume the class slides at the cereal decision rule. We will use the binomial theorem to check that binomial probabilities sum to one, set the success probability to one-half to connect the formula to counting, compare sampling with and without replacement, and introduce Poisson counts with a rain example. The combined notes include additional derivations and examples for review beyond what we will cover in class. The quiz reviews the earlier material through cereal.
A guide to the notes
Select a topic for its PDF pages, or Animate for the matching detailed explanation.
| Topic | Pages / animation | What to focus on |
|---|---|---|
| Independence, covariance, and variance of a sum | 3–8 · Animate | Where independence is used, why covariance appears, and why zero covariance does not establish independence. |
| A single Bernoulli trial | 8–9 · Animate | Code an outcome as zero or one, then derive its expectation and variance. |
| Binomial trials, counting, and normalization | 9–14 · Animate | Count success positions, assign probabilities to sequences, and sum the pmf to one. |
| The cereal decision rule | 14–17 · Animate | Keep the rule fixed while changing the true prize rate. Wednesday resumes at part (b), page 16. |
| A binomial count as a Bernoulli sum | 17–18 · Animate | Linearity gives the expectation; independence removes covariance terms from the variance. |
| Fair-coin counting and sampling with replacement | 18–20 · Animate | Set p = 1/2 to get C(n,k)/2^n, then compare the counting assumptions. |
| Hypergeometric probabilities and moments | 20–25 · Animate | Distinguish population N, sample n, population successes K, and sample successes k. See why each draw has success probability K/N and where dependence enters the variance. |
| Hypergeometric convergence and simulation | 25–28 · Animate | Hold n fixed while the population grows. Compare exact probabilities and 100,000 simulated samples for each population. |
| Poisson: example, pmf, and normalization | 28–29 · Animate | Start with many rare opportunities; identify the expected count and check that the probabilities sum to one. |
| Poisson derivation, exposure, and moments | 29–34 · Animate | Keep np finite in the binomial limit, scale the observation window, and derive mean and variance. |
| Final insurance example and model comparison | 34–35 · Animate | Infer the Poisson parameter from a probability ratio and select models from their assumptions. |
Ideas to carry forward
Expectation is linear without independence. Variance includes covariance terms. Independent Bernoulli trials make those terms zero; sampling without replacement introduces a finite-population correction. Each draw is still a Bernoulli success/failure variable with probability K/N before earlier outcomes are revealed. The notes explain this with a shuffled row and the law of total probability before introducing the term “marginal probability.”
Choose the experiment before the formula. Independent binary trials with a common success probability give a binomial count. A uniform sample without replacement from a fixed population gives a hypergeometric count. Many rare independent opportunities motivate Poisson.
Keep track of what stays fixed. For the hypergeometric-to-binomial limit, the sample size stays fixed while the population grows. For the binomial-to-Poisson limit, the number of opportunities grows, the success probability shrinks, and their product approaches the desired expected count.
Simulation illustrates the result. The figures distinguish exact pmfs from simulated frequencies; the notes also give the mathematical convergence argument.