The language of probability

A trial is a repeatable process that gives one result from a specified set of possibilities. Each possible result is an outcome. Outcomes are equally likely if they all have the same theoretical probability of occurring.

A sample space contains every possible outcome of a trial. It is denoted by $U$, where $U$ is the universal set of outcomes (no unit). One individual outcome within this space is a sample point. An event is a subset of the sample space containing the outcomes of interest.

A sample space can be shown as a list, table or diagram. Lists work well for a small one-stage trial, whereas tables are usually clearer when two actions are performed.

Sample space for two coin tosses

Theoretical and experimental probability

If all outcomes in $U$ are equally likely, the theoretical probability of event $A$ is

Therefore $0 \le P(A) \le 1$.

Relative frequency gives an experimental estimate of probability. It is the proportion of recorded trials in which an event occurs:

Theoretical probability is based on a mathematical model; relative frequency is based on observations. The two don’t have to agree exactly. As the number of independent trials becomes large, relative frequency will often settle near the theoretical probability, provided the model is appropriate. This difference can be seen through repeated physical experiments or computer simulations using spinners, number generators or shuffled objects.

When listing every outcome isn’t practical, a simulation can estimate the probability instead. In a Monte Carlo method, a computational procedure estimates a quantity from relative frequencies by running many random trials. Increasing the number of trials usually reduces random fluctuation, but it cannot fix a biased simulation or an unsuitable model.

Complementary events

The complement of $A$ contains every outcome in $U$ that is not in $A$. It is written $A'$, where $A'$ is the complement of event $A$. Exactly one of $A$ and $A'$ must occur, so

$$P(A')=1-P(A)$$

where $P(A')$ is the probability that event $A$ does not occur (no unit).

Complements are especially useful when “at least one” is easier to find by calculating one minus “none”.

Expected number of occurrences

When the same probability applies across many opportunities, the expected number of occurrences is

An expected count is a long-run average. It doesn’t have to be a possible whole-number result for one group. For example, if $75$ devices each have probability $0.08$ of failing inspection, the expected number failing is $75(0.08)=6$. Exactly six failures are not guaranteed.

Applications and interpretation

Probability models help insurers set prices using estimated lifespans and risks. Governments also use them to plan for likely future demand. In biology, theoretical genetics uses probability structures such as Punnett squares. In physics, a probability model may describe an uncertain particle position.

Gambling presents an ethical issue: a mathematically predictable long-run advantage can earn a profit for an operator, while individual players may respond emotionally to rare rewards. The St Petersburg paradox shows that a game can have a very large theoretical expectation even though people give it a much smaller practical value. Historical work by Russian mathematicians including Chebyshev and Pavlovsky also helped develop the mathematical treatment of uncertainty.

A model and its evidence link theoretical and experimental probabilities; they don’t agree automatically. Reliability depends on assumptions such as equal likelihood, independence and representative trials. Perceived risk isn’t purely mathematical either. People may judge identical probabilities differently in business, medicine and travel because the consequences and emotional responses differ.

Choosing a representation

Many probability problems can be solved straight from a diagram or table, without choosing a formula first.

• A Venn diagram shows events as regions within the sample space. It works well when events overlap or complements are involved.
• A tree diagram uses branches to show successive stages and their conditional probabilities. Multiply probabilities along each route, then add the probabilities of distinct routes.
• A sample space diagram organizes the paired outcomes of two trials.
• A table of outcomes arranges outcomes or frequencies in rows and columns, classified by two variables.

For two events, a Venn diagram divides the sample space into “$A$ only”, “both”, “$B$ only” and “neither”. Fill in the overlap first. If you don’t, it’s easy to count it twice.

Union, intersection and mutually exclusive events

The union of two events occurs when at least one event occurs. The intersection occurs when both do. The addition rule is

In probability, the everyday word “or” is usually non-exclusive, so it includes the overlap. Adding $P(A)$ and $P(B)$ counts the overlap twice, which is why $P(A\cap B)$ must be subtracted.

Mutually exclusive events cannot occur in the same trial. Therefore,

$$P(A\cap B)=0$$

Mutual exclusivity means that occurring together is impossible. It is not the same as independence.

Conditional probability

Conditional probability gives the probability of an event once known information has restricted the sample space. For $P(B)>0$,

Rearrange this to get the multiplication rule:

$$P(A\cap B)=P(B)P(A\mid B)$$

In a tree diagram, each later branch probability is conditional on the route already taken. To find the probability of a complete route, multiply its branch probabilities.

Independence

Independent events are events where knowing that one has occurred doesn’t change the probability of the other. They satisfy

$$P(A\cap B)=P(A)P(B)$$

Equivalently, provided the relevant conditioning probability is non-zero, independence means $P(A\mid B)=P(A)$. Events shouldn’t be treated as independent simply because they seem unrelated. Check the probabilities or use the information given in the context.

Mutually exclusive events with positive probabilities cannot be independent. Once one occurs, the other is impossible, so the first event changes the probability of the second.

With and without replacement

If an item is selected with replacement, it is returned before the next selection. This restores the composition of the collection. Repeated selections are therefore usually independent, and the branch probabilities stay unchanged.

Without replacement, each selection changes the collection. Later probabilities depend on earlier outcomes. A tree diagram is particularly useful in this situation: update both the numerator and denominator on every second-stage branch.

Responsible use

Casinos and other gambling businesses combine probabilities to create a long-run advantage. Mathematics can reveal that advantage and could be used to increase gambling income, but the calculation alone cannot decide whether exploiting players is ethical. Designers, operators, regulators and mathematicians may share responsibility. The numerical model describes likely outcomes; ethical judgement concerns how people use that knowledge.

States and transitions

A discrete dynamical system updates its condition at separate steps using a specified rule. Each possible condition of the system at a given step is called a state.

A Markov chain is a discrete stochastic process where the probability of the next state depends only on the current state, not on the route taken to get there. This “memoryless” assumption needs to fit the context. If earlier history affects future behaviour, the model is unsuitable unless the states are redefined.

A transition probability gives the conditional probability of moving from one state to another in a single step. In a transition diagram, nodes represent states and directed arrows show transitions, labelled with their probabilities. A loop shows that the system remains in the same state.

Transition matrices

A transition matrix is a square matrix containing one-step transition probabilities. Under the column convention used in the course,

Read each transition from column $j$ to row $i$. A column lists every possible destination from one current state. Each entry therefore lies from $0$ to $1$, and every column sums to $1$.

A state probability matrix is a column matrix listing the probabilities of occupying each state at a particular step. All entries are non-negative and add to $1$. The system evolves according to

For example, $s_1=Ts_0$ and $s_2=T^2s_0$. The multiplication order matters and can’t be reversed. The power $T^n$ has a direct meaning too: its entry in row $i$, column $j$ gives the probability of reaching state $i$ from state $j$ in exactly $n$ transitions.

Regular chains

A regular Markov chain has a transition matrix for which some positive integer power contains only positive entries. Practically, this means that after enough steps, every state can be reached from every starting state with positive probability.

A regular chain has stable long-term behaviour. Its state probabilities approach the same steady state, whatever the initial probability matrix. A chain with an absorbing state is not generally regular, since that state cannot be left.

Finding the steady state

A steady-state matrix is a state probability matrix that stays unchanged after one transition. Denoting it by $s$

Two approaches are required.

1. Repeated multiplication: calculate $s_1,s_2,s_3,\ldots$ until corresponding entries stabilize to the required accuracy. This produces numerical long-term probabilities.
2. Simultaneous equations: write out $Ts=s$ alongside the condition that every entry of $s$ sums to $1$, then solve. Use this approach when an exact solution is required.

A technology-generated table of successive state probabilities shows convergence clearly. However, apparent convergence doesn’t prove that a chain is regular.

Successive state probabilities approaching a steady state.

Since $Ts=s$ can also be written as $Ts=1s$, the steady state is an eigenvector of $T$ corresponding to eigenvalue $1$, scaled so that its entries sum to $1$. This connects transition matrices with matrix multiplication, eigenvalues and adjacency-style representations of directed networks.

Further contexts

An absorbing state can’t be left once it has been entered, so its self-transition probability is $1$. Absorbing chains may model eventual removal, failure or completion. One example is the gambler’s ruin model, where a player’s fortune changes step by step until a boundary fortune is reached and becomes absorbing.

Computer simulation can produce many individual paths through a Markov chain, allowing long-term or absorption probabilities to be estimated. By contrast, repeated matrix multiplication tracks the entire probability distribution directly.

Similar matrix models occur in biology. Leslie matrices, for instance, project age-structured populations. Their entries contain survival and reproduction rates, however, rather than forming an ordinary transition matrix whose columns must all sum to $1$.