Probability quantifies uncertainty inside a model you state. Statistics uses data to estimate, test, or update a model. The two subjects share a vocabulary and ask different questions. Probability asks what should happen if the model is true. Statistics asks what the data suggest about the world, and how fragile that suggestion is. This pillar is a first course map of that distinction, with a study order into the Bayes explainer, the Bayes theorem calculator, and the other probability tools.

Who this pillar is for

Start here if you are beginning a first course in chance or data, or if you can compute a fraction and still mix up $P(A\mid B)$ with $P(B\mid A)$. You might be reading a result that says "significant" and wanting to know what was actually claimed. You might be about to meet the binomial and normal models and wanting a sentence for each before a calculator prints a tail.

This site does not have a separate probability pathway. The pathways hub orders calculus, linear algebra, and proofs. Use this page as the cluster. If a later probability course leans on integrals, repair accumulation on calculus foundations when you get there, not before the counting makes sense.

Prerequisites

You need fractions, percentages, and the patience to define the experiment before you compute. You should be able to list a small set of outcomes and count them. Algebra is enough for the pages linked here. Calculus becomes useful when densities and continuous expectations arrive. It is not a gate for conditional probability.

Read probability, conditional probability, random variable, expected value, and Bayes theorem when you want the short form. OpenIntro Statistics is on the sources page if you want a full open book beside this map.

What you should be able to do

When you finish the core, you should be able to do these things without hiding behind a phrase.

  • State the sample space and the event before you assign a number.
  • Compute a conditional probability by restricting attention to the outcomes inside the condition.
  • Say what independence claims: learning one event does not change the probability of the other. It is a modeling assumption.
  • Update in the Bayes direction, from $P(E\mid H)$ toward $P(H\mid E)$, on a small counting example.
  • Tell a binomial count from a normal model, and say what a expected value averages.
  • Read a statistical claim as an argument with assumptions, including what a p value does and does not say.

Events, conditioning, and independence

An event is a set of outcomes you have decided to care about. Conditional probability $P(A\mid B)$ restricts attention to outcomes inside $B$. If $P(B)$ is positive, $P(A\mid B)=P(A\text{ and }B)/P(B)$.

Independence means $P(A\mid B)=P(A)$ when $P(B)$ is positive. Learning $B$ did not move the probability of $A$. Fair coin flips are often modeled that way. If two lights share a switch, independence is false even when each light is on about half the time. The margins can look ordinary while the joint behavior is tied.

Counting picture. Two jars. Jar 1 has 3 red marbles and 1 blue marble. Jar 2 has 1 red and 3 blue. You pick a jar at random, then a marble at random. Each jar has probability $1/2$. Given that the marble is red, the probability it came from jar 1 is $3/4$, not $1/2$. The red marble is three times as likely from jar 1 as from jar 2, and the jars were equally likely before you looked. That arithmetic is Bayes theorem on four equally likely jar draws, scaled by the color counts. Bayes theorem without the fog tells the same story with a larger hall and a medical test, and it warns about the false positive rate.

Bayes theorem as a change of direction

A test is often described by $P(\text{evidence}\mid \text{hypothesis})$: how often the evidence appears when the hypothesis is true. The question people mean is $P(\text{hypothesis}\mid \text{evidence})$. Bayes theorem turns one into the other once you have a prior and the probability of the evidence overall.

In the jar picture the two conditionals both equal $3/4$. That match is an accident of the numbers. Empty jar 2 of red marbles and a red draw must have come from jar 1. Make jar 2 much more common as a prior, and $P(\text{jar 1}\mid \text{red})$ drops even though jar 1 is still better at producing red. The explainer's lecture hall is that lesson with a rare condition. Read it after the jar count.

When the prior, the likelihood, and the false positive rate are numbers you trust, the Bayes theorem calculator plugs them into the formula and shows the two pieces that add up to the probability of the evidence. Predict the posterior from a count table before you press Compute. The tool will not notice a swapped conditional.

Random variables, binomial counts, and the normal model

A random variable assigns a number to each outcome. Its distribution says how those numbers are spread. Expected value is the probability weighted average, a center of mass. It need not be a value you can actually draw. A fair six sided die has expected value $3.5$. You will never roll $3.5$. The number is the long run average of many rolls under the model, not a promise about one roll.

A binomial count is the number of successes in a fixed number of independent trials with the same success probability $p$. For a fair coin, five tosses, the chance of exactly 3 heads is $\binom{5}{3}/32 = 10/32$. Count the binomial coefficient on the combinatorics calculator if that step is rusty, then confirm the probability on the binomial calculator. The combinatorics page only counts. It does not know $p$.

A normal model is a bell shaped curve with a mean and a standard deviation. Standardizing, $z=(x-\mu)/\sigma$, asks how many standard deviations $x$ sits from the mean. A score of $130$ with mean $100$ and standard deviation $15$ has $z=2$. About $0.023$ of a standard normal sits above $2$. The normal CDF calculator reports a tail. Name the upper or lower tail before you read the digits.

Statistics: estimates with the assumptions in view

A sample mean estimates a population mean when the sample matches what the formula assumes. A 95 percent confidence procedure, under the model, covers the parameter in about 95 percent of repeated samples. That is a statement about the method. Reading one finished interval as "the parameter has probability 95 percent of sitting here" is a different question, and it needs a prior.

A hypothesis test measures surprise under a stated model. It does not, by itself, measure importance. A tiny effect in a huge sample can look surprising. A large effect in a tiny sample can fail to. Ask what was sampled, what else could explain the pattern, and whether the model was plausible before the test.

Suggested learning order

Work in this order. One model per sitting is enough. Open a calculator after you can name the sample space or the distribution.

  1. Read probability, conditional probability, and Bayes theorem. Compute the two jar update by counting before you read further.
  2. Read Bayes theorem without the fog. Keep $P(E\mid H)$ and $P(H\mid E)$ on two separate lines of your notes. Then run the same inputs on the Bayes theorem calculator.
  3. Read random variable and expected value. Compute the expected value of a fair die by hand. You should get $3.5$.
  4. Use the combinatorics calculator for $\binom{5}{3}$, then the binomial probability calculator for 3 successes in 5 fair trials. Predict $10/32$ first.
  5. Standardize one number by hand, then use the normal CDF and z score calculator for the tail you named in advance.
  6. If densities and integrals are the next chapter, read calculus foundations and practice accumulation on the Riemann sum calculator. For a full book, open the statistics title on the sources page.

Checkpoints and practice habits

Practice habit. Before any probability, write the experiment in one sentence: what is random, what is fixed, and what you are conditioning on. If the sentence is missing, the fraction is not a probability yet. It is only arithmetic.

Bayes check. In the two jar story, $P(\text{jar 1}\mid \text{red})=3/4$. If you got $1/2$, you ignored the color counts. If you got $3/4$ for the wrong conditional, write both $P(\text{red}\mid \text{jar 1})$ and $P(\text{jar 1}\mid \text{red})$ and see that the match was luck. The Bayes theorem calculator should print $3/4$ for prior $1/2$, likelihood $3/4$, and $P(E\mid \text{not }H)=1/4$.

Binomial check. Five fair coin tosses, exactly three heads: $10/32$. The binomial calculator should agree. If it does not, check that you asked for exactly three successes, not "at least three."

Expectation check. A prize of $0$ with probability $0.9$ and $100$ with probability $0.1$ has expected value $10$. Most draws pay nothing. The expected value is not a typical single draw when the distribution is lopsided.

Common pitfalls

Pitfall. Mixing up $P(A\mid B)$ and $P(B\mid A)$ is the classic error, and a pretty test makes it worse when the condition is rare. "Statistically significant" is a technical phrase about a test under assumptions. It is not a synonym for important. A p value is computed by assuming a model and asking how surprising the data look. It is a different number from the probability that the hypothesis is true.

A calculator will multiply the binomial formula you hand it. It will not notice that the trials were dependent, or that $n$ was not fixed, or that you wanted a tail and asked for a point. The FAQ on calculators versus understanding applies here as much as in calculus. Independence is a modeling assumption. Write it down or do not use a formula that needs it.

Related reading

Stay with Bayes theorem without the fog until the two conditionals feel different. Check a posterior on the Bayes theorem calculator. Counting practice is the combinatorics, binomial, and normal calculators. Proof habits transfer from real analysis when a course asks for them. A full open statistics book is on the sources page.

Do the two jar count on paper, then read the Bayes explainer. Open a calculator only after the model is named.

Read Bayes theorem without the fog
Disclaimer. Educational only. This page is not statistical advice, not a data analysis, and not a substitute for a course. See the disclaimer.