Science & Data

Probability

Write a probability with P(...). The value must be between 0 and 1, and a percentage works too, so P(10%) is the same as P(0.1). A probability is just a number, so ordinary arithmetic still applies.

P(0.1)
P(10%)
P(0.3) - P(0.1)

Function names are case-insensitive, so P(0.5) and p(0.5) are the same.

Combining probabilities

Use the bitwise operators on probabilities. They switch to probability math whenever at least one operand is a P(...).

Expression	Meaning	Result
`~P(0.2)`	Complement (NOT), `1 − p`	`0.8`
`P(0.5) & P(0.3)`	Intersection (AND), `p·q`	`0.15`
`P(0.5) \| P(0.3)`	Union (OR), `p + q − p·q`	`0.65`
`P(0.5) xor P(0.3)`	Exactly one, `p + q − 2·p·q`	`0.5`

P(0.1) & P(0.2)
P(0.1) | P(0.2)
~P(0.2)
P(0.2) & P(0.3) & P(0.1)

& binds tighter than | (AND before OR), and combinations stay probabilities, so they chain.

These formulas assume the events are independent. For mutually exclusive events the union is simply P(a) + P(b) — add them directly instead of using |.

Functions

Function	Description	Example
`conditional(pAandB, pB)`	Conditional probability P(A\|B)	`conditional(0.12, 0.3)`
`bayes(prior, likelihood, fp)`	Bayes' posterior P(H\|E)	`bayes(0.01, 0.9, 0.05)`
`odds(p)`	Odds for a probability, `p / (1 − p)`	`odds(0.75)`
`probability from odds(o)`	Probability implied by odds, `o / (1+o)`	`probability from odds(3)`
`binomial(n, k, p)`	k successes in n independent trials	`binomial(10, 3, 0.5)`
`expected value(v1, p1, …)`	Weighted sum of value/probability pairs	`expected value(10, 0.5, 20, 0.5)`

bayes takes the prior P(H), the likelihood P(E|H), and the false-positive likelihood P(E|¬H):

bayes(0.01, 0.9, 0.05)
binomial(10, 3, 0.5)
expected value(100, 0.3, 0, 0.7)

Common mistakes

P(50) is invalid — a probability is between 0 and 1. Write P(50%) or P(0.5).
P(0.5) | P(0.3) is union (OR), not "P of A given B". Conditional probability is the separate conditional(...) function.
Without any P(...), & and | stay integer bitwise operators, so 5 & 3 is still a bitwise AND.
Values below 1 are shown to 3 significant figures, so binomial(10, 3, 0.5) prints 0.117.