Probability Distributions Cheat Sheet — GetCalcMaster
A practical guide to common probability distributions: when to use normal, binomial, Poisson, exponential, and how to sanity-check results.
A practical cheat sheet for common distributions
Probability questions often look different on the surface but share the same underlying distributions. The fastest way to solve them is to recognize the pattern, pick the right distribution, and sanity‑check the result.
This guide is intentionally practical: it focuses on when to use what, and the mistakes that make answers wrong.
Binomial: fixed number of trials
Use binomial when you have:
- a fixed number of trials
n - each trial is success/failure
- success probability
pis constant
Example: “In 10 coin flips, what is the probability of exactly 3 heads?”
Try it: binompmf(3,10,0.5)
Poisson: counts in a time/space window
Use Poisson when you have counts of rare events over a fixed interval with an average rate λ:
- arrivals per hour
- defects per meter
- calls per day
A key sanity check: the mean and variance are both λ.
Normal: sums of many small effects
Use normal when you’re modeling measurement noise or sums/averages of many small independent effects. Normal is also used as an approximation to binomial when n is large and p isn’t extreme.
Try a CDF sanity check: normalcdf(0) should be 0.5 for a standard normal.
Exponential: waiting time between Poisson events
If events follow a Poisson process with rate λ, then the waiting time between events is exponential. The mean waiting time is 1/λ.
Uniform: “equally likely” within a range
Use uniform only when you have a reason to believe every value in an interval is equally likely. It’s often a weak assumption used for bounding.
Common mistakes (that change the distribution)
- Changing p across trials breaks binomial assumptions.
- Dependent trials (sampling without replacement) may require hypergeometric instead of binomial.
- Confusing rate and probability: λ is a rate; p is a probability.
- Forgetting units: λ must match the interval (per hour vs per day).
GetCalcMaster workflow for probability problems
- Write down what is fixed: n? time window? rate?
- Pick the distribution family.
- Compute the quantity you need (pmf/cdf/inv).
- Sanity check (bounds, mean/variance, symmetry).
- Store the calculation in Notebook with a one-line explanation.
FAQ
How do I choose between binomial and Poisson?
Use binomial for a fixed number of trials with probability p. Use Poisson for counts over an interval with average rate λ (especially when events are rare).
When is normal approximation reasonable?
As a rule of thumb, normal approximation to binomial is better when n is large and both np and n(1-p) are not too small. Always sanity-check against exact binomial when possible.
Is probability output “exact”?
It’s computed numerically. For many distributions, results are approximations to real values. For learning and typical applications, this is fine — just verify when stakes are high.