Introduction to Probability

Begin with an experiment, an outcome and an event

An experiment is a procedure with a result that is uncertain before it is observed. An outcome is one fully specified result at the level of detail being recorded. The sample space, written $S$, contains all such outcomes. An event is a subset of that sample space.

For one roll of a six-sided die,

The event $A$ that the result exceeds four is $A=\{5,6\}$. If the die is fair, each face has probability $1/6$, so

The braces identify sets. The number 5 is an outcome; the set $\{5\}$ is the event containing only that outcome. In ordinary calculation, $P(5)$ is often used as a convenient shorthand.

Probability has a scale and a model

Probabilities satisfy

Here $\varnothing$ is the empty event. Under the fair-die model, rolling seven has probability zero because it is not in the sample space. Rolling a number from one to six has probability one. A probability of $1/2$ says an event and its complement have equal probabilities; it does not promise alternation between them.

Fractions, decimals and percentages are representations of the same probability. For example, $1/4=0.25=25\%$. An answer of $1.4$ or $140\%$ cannot be a probability. These bounds give a quick check, although a result inside the bounds can still be wrong.

For these finite equally likely examples, an event with no outcomes has probability zero. Later, Probability Distributions explains why a particular exact value can have probability zero in a continuous model without being logically impossible.

Count outcomes only when they are equally likely

For a finite sample space whose elementary outcomes are equally likely,

The notation $|A|$ means the number of outcomes in $A$. Equal likelihood is a condition of this formula, not a consequence of listing outcomes. A label on a spinner does not make its chance equal to that of every other label.

Consider a spinner model with three sectors having probabilities $1/2$, $1/3$ and $1/6$. The chance of either of the last two sectors is $1/3+1/6=1/2$, not $2/3$. There are two acceptable labels out of three, but those labels do not carry equal probability. The probabilities, rather than just the labels, must be added.

Probability terminology and equal-likelihood calculations provide the foundation for this distinction. The examples on this page use stipulated models, not measured CTS outcomes.

Two coin tosses: four outcomes, three counts

Suppose two tosses are independent and each has probability $1/2$ of heads. Record the first toss before the second:

Each ordered result has probability $1/4$. Exactly one head can occur as $HT$ or $TH$, so its probability is $2/4=1/2$. Two heads occur only as $HH$, with probability $1/4$.

The number of heads can be 0, 1 or 2. Those three counts are not equally likely: their probabilities are $1/4$, $1/2$ and $1/4$. Grouping elementary outcomes into counts preserves their probability totals, not equal weights. This is the first step towards a random variable.

The independence condition also matters. Two individual tosses could each have a 50% heads probability but be perfectly linked, with the second always matching the first. In that different model, only $HH$ and $TT$ occur. Individual probabilities alone do not specify the joint experiment.

From a model to a calculation

A useful solution states the experiment and assumptions, identifies the requested event, assigns outcome weights, and then combines the relevant weights. Finish by interpreting the result in the language of the situation.

For example, a bag contains three tokens marked A and two marked B. Suppose one token is chosen so that every individual token is equally likely. There are five elementary token outcomes, three of which have an A label. Therefore $P(A)=3/5=0.6$. The result describes this selection procedure. It does not mean that any physical grab from the bag, however performed, must have the same probability.

If two tokens will be drawn, the procedure needs more detail: is the first token replaced, and is the next selection freshly randomized? Those questions change the later probabilities. The Multiplication Rule of Probability develops both models.

Probability and observed frequency

A model probability is not the same object as the proportion observed in one set of trials. If heads appear seven times in ten tosses, the observed proportion is $7/10$. That does not automatically establish that the coin’s heads probability is $0.7$, nor does it force tails to occur next.

Repeated observations can help assess a model, but the design matters. Stable conditions and suitable independence assumptions support the usual long-run interpretation. Theoretical and Experimental Probability examines why relative frequencies fluctuate and why a short streak is not a guarantee about the next outcome.

Related topics

Use Sample Spaces and Events to organise more complicated outcomes. Then study Theoretical and Experimental Probability to connect a model with observations.