Bayesian Statistics Basics Cheat Sheet

A printable reference covering Bayes’ theorem, priors, likelihoods, posteriors, evidence, and Bayesian updating for grades 11-12.

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Bayesian statistics is a way to update probabilities when new evidence is observed. This cheat sheet helps students connect conditional probability to real statistical reasoning. It is especially useful for interpreting test results, predictions, and uncertain claims using evidence. Students need it to see how prior beliefs and data combine in a clear mathematical process. The central formula is Bayes’ theorem, $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$ . In Bayesian thinking, the prior probability represents what is believed before seeing the data, the likelihood measures how compatible the data are with a hypothesis, and the posterior probability is the updated belief. The denominator, often called evidence, makes the probabilities add to $1$ . Bayesian updating can be repeated as new data arrive.

Key Facts

Bayes’ theorem is $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$ , where $P(A\mid B)$ is the updated probability of event $A$ after observing event $B$ .
The prior probability is $P(H)$ , which represents the probability of a hypothesis $H$ before observing new data.
The likelihood is $P(D\mid H)$ , which represents the probability of observing data $D$ if hypothesis $H$ is true.
The posterior probability is $P(H\mid D)=\frac{P(D\mid H)P(H)}{P(D)}$ , which represents the updated probability after observing data $D$ .
The evidence can be found by total probability: $P(D)=P(D\mid H)P(H)+P(D\mid H^c)P(H^c)$ for a hypothesis $H$ and its complement $H^c$ .
Posterior odds are prior odds multiplied by the likelihood ratio: $\frac{P(H\mid D)}{P(H^c\mid D)}=\frac{P(H)}{P(H^c)}\cdot\frac{P(D\mid H)}{P(D\mid H^c)}$ .
If events $A$ and $B$ are independent, then $P(A\mid B)=P(A)$ , so observing $B$ does not change the probability of $A$ .
Bayesian updating can be repeated because today’s posterior probability can become tomorrow’s prior probability.

Vocabulary

Prior probability: The probability assigned to a hypothesis before considering the new data.
Likelihood: The probability of observing the data assuming a particular hypothesis is true.
Posterior probability: The updated probability of a hypothesis after using Bayes’ theorem with the observed data.
Evidence: The overall probability of the observed data across all possible hypotheses.
Conditional probability: The probability that one event occurs given that another event has already occurred.
Likelihood ratio: A comparison of how likely the data are under one hypothesis versus another hypothesis.

Common Mistakes to Avoid

Confusing $P(A\mid B)$ with $P(B\mid A)$ is wrong because the condition changes the group being considered, and the two probabilities are usually not equal.
Ignoring the prior $P(H)$ is wrong because Bayesian statistics always combines prior information with the likelihood from new data.
Forgetting to divide by the evidence $P(D)$ is wrong because the posterior probabilities must be normalized so they add to $1$ across all hypotheses.
Treating a high test accuracy as the same as a high posterior probability is wrong because the base rate, or prior probability, can strongly affect $P(H\mid D)$ .
Using $P(D)=P(D\mid H)+P(D\mid H^c)$ is wrong because each likelihood must be weighted by its prior, giving $P(D)=P(D\mid H)P(H)+P(D\mid H^c)P(H^c)$ .

Practice Questions

1 A disease affects $2\%$ of a population. A test is positive $95\%$ of the time for people with the disease and $10\%$ of the time for people without it. Find $P(\text{disease}\mid \text{positive})$ .
2 A factory machine makes $30\%$ of all parts, and $4\%$ of its parts are defective. The other machine makes $70\%$ of all parts, and $1\%$ of its parts are defective. Find the probability a defective part came from the first machine.
3 A hypothesis has prior probability $P(H)=0.25$ . The data have likelihoods $P(D\mid H)=0.8$ and $P(D\mid H^c)=0.2$ . Find $P(H\mid D)$ .
4 Explain why a rare condition can still have a low posterior probability after a positive test result, even if the test is fairly accurate.

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