Probability

Last updated: 2026-04-03

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Rmd	e8c428a	Dave Tang	2026-04-03	Fix LaTeX
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Rmd	555e3e3	Dave Tang	2026-04-03	Bayes’ theorem
html	c17bafb	Dave Tang	2026-04-03	Build site.
Rmd	73835bb	Dave Tang	2026-04-03	Marginal and conditional probability
html	902bbc7	Dave Tang	2026-01-07	Build site.
html	fe7d8b8	Dave Tang	2023-11-24	Build site.
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html	97b7d02	Dave Tang	2023-06-27	Build site.
html	e08ce00	davetang	2020-10-14	Build site.
Rmd	805d205	davetang	2020-10-14	Probability basics

A lot of bioinformatics tools use probabilistic models, so it’s important to have at least a working understanding of probability. In simpler cases, if you can figure out all the possible events, you can calculate the probability of a subset of events by dividing the subset by all possible events. For example, in a full deck of cards, there are 52 cards and the probability of picking a red card is the subset of red cards (26) divided by all the cards (52).

Another way to think about the probability of an event is that it is the proportion of times that the event occurs when we repeat the experiment an infinite number of times, independently and under the same conditions. The probability is the number of times an event occurred divided by the total number of times an experiment was repeated. Because of this, we can get a estimate of the probability of an event by running Monte Carlo simulations. This is particularly useful when it is difficult to work out the probability mathematically.

Basics

\(Pr(A)\) is the probability of event \(A\) happening; the general term event is used to refer to things that can happen by chance.
A probability distribution is the probabilities of all possible events.
For discrete variables, the probability distribution is calculated using a probability density function (PDF).
For continuous variables, the probability distribution is calculated using the cumulative distribution function (CDF).
Two events are independent if the outcome of one does not affect the other. The classic example is coin tosses; every time we toss a fair coin, the probability of seeing heads is 1/2 regardless of what previous events were.

When events are not independent, conditional probabilities are used.

\[ Pr(Card\ 2\ is\ a\ king\ |\ Card\ 1\ is\ a\ king) = \frac{3}{51} \]

The \(|\) means “given that” or “conditional on”. The mathematical definition of independence is:

\[ Pr(A\ |\ B) = Pr(A) \]

The fact that event \(B\) has occurred does not affect the probability of \(A\).

We use the multiplication rule for calculating the probability of two events occurring:

\[ Pr(A\ and\ B) = Pr(A) \cdot pr(B\ |\ A) \]

When there are three events:

\[ Pr(A\ and\ B\ and\ C) = Pr(A) \cdot Pr(B\ |\ A) \cdot Pr(C\ |\ A\ and\ B) \]

If the events are independent, then:

\[ Pr(A\ and\ B\ and\ C) = Pr(A) \cdot Pr(B) \cdot Pr(C) \]

General formula for computing conditional probabilities:

\[ Pr(B\ |\ A) = \frac{Pr(A\ and\ B)}{Pr(A)} \]

If event \(A\) and \(B\) are independent, then the probability just becomes \(Pr(B)\):

\[ Pr(B\ |\ A) = \frac{Pr(A) \cdot Pr(B)}{Pr(A)} = Pr(B) \]

Addition rule for calculating the probability of either event happening (but not both, which is why we subtract):

\[ Pr(A\ or\ B) = Pr(A) + Pr(B) - Pr(A\ and\ B) \]

Bayes’ theorem:

\[ Pr(A\ |\ B) = \frac{Pr(B\ |\ A) \cdot Pr(A)}{Pr(B)} \]

Again, if event \(A\) and \(B\) are independent, then the probability just becomes \(Pr(B)\):

\[ Pr(A) = \frac{Pr(B) \cdot Pr(A)}{Pr(B)} \]

One application of Bayes’ Theorem is for spam filtering where:

Event A: the message is spam
Text: the message contains certain words

\[ Pr(spam\ |\ words) = \frac{Pr(words\ |\ spam) \cdot Pr(spam)}{Pr(words)} \]

Marginal and Conditional Probability

A 2 by 2 contingency table can help illustrate marginal and conditional probability.

disease_table <- matrix(
  c(40, 10, 20, 30),
  nrow = 2,
  dimnames = list(
    "Test" = c("Test +", "Test -"),
    "Disease" = c("Disease +", "Disease -")
  )
)

disease_table_with_margins <- addmargins(disease_table)

disease_table_with_margins

        Disease
Test     Disease + Disease - Sum
  Test +        40        20  60
  Test -        10        30  40
  Sum           50        50 100

Marginal probabilities are read directly from the totals in the margins of the table, ignoring the other variable. Reading the Sum, probability of disease (Disease +) is 0.5 and probability to test positive (Test +) is 0.6.

disease_table |>
  prop.table() |>
  addmargins()

        Disease
Test     Disease + Disease - Sum
  Test +       0.4       0.2 0.6
  Test -       0.1       0.3 0.4
  Sum          0.5       0.5 1.0

Conditional probabilities restrict attention to a single row or column. Given a positive test, we restrict ourselves to the first row.

\(P(Disease+|Test+)\) is of those who tested positive, what fraction have disease?

\[ P(Disease+|Test+) = 0.4 / 0.6 = 0.67 \]

Given positive to disease, we restrict ourselves to the first column.

\(P(Test+|Disease+)\) is of those with disease, what fraction test positive?

\[ P(Test+|Disease+) = 0.4 / 0.5 = 0.80 \]

Note that \(P(Disease+|Test+)\) is not equal to \(P(Test+|Disease+)\); conditional probability is not symmetric. This asymmetry is the source of the base rate fallacy, which is commonly encountered in medical testing and legal reasoning, where \(P(Disease|Test+)\) is confused with \(P(Test+|Disease)\).

Bayes’ theorem

A cell in the two by two contingency table is simply a tally, a count of people who satisfy two conditions simultaneously. The top-left cell is how many people in our total of 100 had both a positive test and the disease.

There are two different ways to express the probability of landing in any single cell of the table.

disease_table |>
  addmargins()

        Disease
Test     Disease + Disease - Sum
  Test +        40        20  60
  Test -        10        30  40
  Sum           50        50 100

Consider the top-left cell with 40 people who are both Test+ and Disease+.

Start with Disease, then look at Test.

If we pull aside everyone with disease, we end up with 50 people. Then from those 50, if we ask how many tested positive we will get 40.

First taking a 50% slice of 100 (Disease+)
Then taking an 80% slice of those (ones who also tested positive)

\[ 50\%×80\%=40\% \]

Start with Test, then look at Disease

Again, if we pull aside everyone who tested positive, we end up with 60 people. Then from those 60, if we ask how many actually have the disease we get 40 again.

First taking a 60% slice of 100 (Test+)
Then taking a 67% slice of those (the ones who also have disease)

\[ 60\%×67\%=40\% \]

is actually the same as

\[ P(Disease+) \times P(Test+∣Disease+) \]

\(P(Disease+)=50/100=50\%\), which is pulling aside Disease+ (marginal)
\(P(Test+∣Disease+)=40/50=80\%\), which is then asking among Disease+, how many tested positive (conditional)

is the same as

\[ P(Test+) \times P(Disease+∣Test+) \]

\(P(Test+)=60/100=60\%\), which is pulling aside Test+ (marginal)
\(P(Disease+∣Test+)=40/60=67\%\), which is then asking among Test+, how many have disease (conditional)

Therefore:

\[ P(Disease+) \times P(Test+∣Disease+) = P(Test+) \times P(Disease+∣Test+) \]

So if we want to know \(P(Disease+ | Test+)\) but only have \(P(Test+ | Disease+)\), we simply rearrange:

\[ P(Test+) \times P(Disease+∣Test+) = P(Disease+) \times P(Test+∣Disease+) \\ P(Disease+∣Test+) = \frac{P(Disease+) \times P(Test+∣Disease+)}{P(Test+)} \]

This is Bayes’ theorem. In the abstract form with A and B:

\[ P(A | B) = \frac{P(B | A) \times P(A)}{P(B)} \]

sessionInfo()

R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.3 LTS

Matrix products: default
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time zone: Etc/UTC
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attached base packages:
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other attached packages:
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Probability

Dave Tang

2026-04-03

Basics

Marginal and Conditional Probability

Bayes’ theorem