01 · Lesson 08

Probability

Probability describes uncertainty before an outcome is known. In mathematical biology it can represent infection, recovery, extinction, mutation, survival or measurement uncertainty.

Scenario: will an introduced infection disappear?

Two communities begin with the same population and disease parameters. Chance infection and recovery events may nevertheless produce different epidemic paths. In one community the infection may disappear early; in another it may produce a major outbreak.

Probability does not say which individual future must occur. It quantifies uncertainty across the outcomes allowed by a stated model.

Sample space and events

The sample space \(\Omega\) is the set of possible outcomes. An event \(A\) is a subset of those outcomes. Examples include “extinction by day 20” or “peak infections exceed 100.”

A probability satisfies

\[0\le P(A)\le1,\qquad P(\Omega)=1.\]

The impossible event has probability zero. Mutually exclusive events cannot occur together.

Complements, unions and intersections

\[P(A^c)=1-P(A).\]

The union \(A\cup B\) means that at least one event occurs. The intersection \(A\cap B\) means both occur. In general,

\[P(A\cup B)=P(A)+P(B)-P(A\cap B).\]

If the events are mutually exclusive, the intersection probability is zero and their probabilities add directly.

Conditional probability

The probability of \(A\) given that \(B\) has occurred is

\[P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0.\]

For example, the probability of hospital-capacity exceedance conditional on a major outbreak differs from the unconditional probability across all epidemics. The conditioning event must always be stated.

Independence

Events \(A\) and \(B\) are independent when

\[P(A\cap B)=P(A)P(B).\]

Independence is a model property, not a default assumption. Epidemic outcomes observed at nearby times are generally dependent, and individuals sharing contacts or environments may also be dependent.

Random variables and distributions

A random variable assigns a numerical value to each random outcome. The final outbreak size \(Z\), epidemic duration \(T\) and peak infectious count \(I_{\max}\) are examples.

A probability distribution specifies which values are possible and how probability is allocated among them. Discrete variables use probability masses; continuous variables use densities, where probability is obtained over an interval rather than at an isolated point.

Expectation and variance

For a discrete random variable \(X\) with possible values \(x\),

\[\mathbb E[X]=\sum_x xP(X=x).\]

Expectation is a probability-weighted average across possible outcomes. It need not itself be a possible outcome. Variance measures spread around the expectation:

\[\operatorname{Var}(X)=\mathbb E[(X-\mathbb E[X])^2].\]

Two epidemic models can have similar expected peak sizes but very different variances and tail probabilities. The mean alone does not describe risk.

Rates and short-interval probabilities

If an event has instantaneous rate \(a\), then for a sufficiently short interval \(\Delta t\),

\[P(\text{one event in }[t,t+\Delta t))=a\Delta t+o(\Delta t).\]

A rate may exceed one because it has units of events per time; a probability cannot. Multiplying by a short time produces an approximate dimensionless probability. This approximation is not the definition of a probability for an arbitrary long interval.

Model probability and estimated probability

A model may define a probability exactly, while simulation estimates it using a finite sample. If \(m\) of \(M\) independent trajectories satisfy event \(A\),

\[\widehat P(A)=\frac{m}{M}.\]

The estimate changes across simulation experiments. Its Monte Carlo uncertainty should be reported rather than treating it as the exact model probability.

Conditions and limitations

Define the experiment, sample space and event before assigning a probability.
Distinguish conditional from unconditional probability.
Do not assume independence without justification.
Separate intrinsic biological randomness from uncertainty about parameter values.
State whether a probability is exact, approximated analytically or estimated by simulation.
A probability model is meaningful only relative to its biological assumptions.

What this lesson adds

You can now define events and random variables, use complements and conditional probability, test the meaning of independence, interpret expectation and variance, and distinguish event rates from probabilities and exact probabilities from simulation estimates.