The bar is a division sign, not a direction arrow
A conditional probability answers a question of the form: if we already know B is true, how likely is A? The standard notation P(A|B) is defined as the probability that both A and B occur, divided by the probability of B [3]. So the left slot holds the event you are asking about, and the right slot holds what you are taking for granted. The vertical bar is best read as 'given,' and the whole expression as 'the probability of A given B.'
This is why the convention is sometimes described as right-to-left in spirit: you read the condition first, then the event it bears on. But the writing order follows the grammar of the ratio, not a rule that conditions must come last. The notation is a convenience, and one paper explicitly treats P(A|B) as shorthand for that ratio rather than as a claim about which event is logically prior [3].
Swapping the two slots can change everything
The real reason the order is drilled into students is that P(A|B) and P(B|A) are generally different numbers. Confusing them is a documented reasoning error called 'flipping the conditional,' and its consequences can be severe [2]. The error is easiest to make when the events involved are very rare, because then the two quantities can diverge dramatically while both feeling intuitively small.
A concrete case is medical testing. The sensitivity of a test — the probability of a positive result given disease — is not the same as its positive predictive value — the probability of disease given a positive result [2]. A test can be highly sensitive and still, when the disease is rare, produce mostly false positives. The same confusion underlies the 'prosecutor's fallacy,' where the probability of the evidence given innocence is mistaken for the probability of innocence given the evidence [2]. In both cases the notation's two slots carry genuinely different information, and the right-hand slot tells you what world you are standing in.
Why this convention rather than the reverse
The supplied evidence treats the left-right placement as a convention rather than a derived law. One paper notes that it follows convention in using P(A|B) as shorthand for the ratio, while urging readers to resist a further convention about how the expression is read aloud [3]. That framing suggests the placement is a shared habit of the field, not something forced by the mathematics.
The convention also fits how conditional probabilities are used in practice. When we reason about a hypothesis in light of evidence, we typically name the hypothesis first and the evidence second, and the ratio definition puts the joint event on top and the conditioning event below [3]. Other work extends conditional probability into quantum modeling with a bracket notation, showing that the underlying idea — a probability that depends on another event being true — travels across formalisms even when the symbols change [5]. The dependence itself is the substance; the bar's position is bookkeeping.
The notation is simple; the reasoning is not
Conditional probability is central to how people evaluate 'if' statements. One account holds that a simple indicative conditional is acceptable to the extent that the corresponding conditional probability is high, provided the condition is relevant to the outcome [4]. That proposal adds a relevance requirement on top of the probability, which hints that the two slots are not interchangeable: the condition has to actually bear on the event for the conditional to make sense.
None of this makes the right-side placement inevitable. The papers that discuss the notation treat it as a convention, and the paper on conditionals and conditional probabilities focuses on when a conditional probability is defined at all rather than on where the bar goes [1]. The practical lesson for a reader is to slow down at the bar: ask which event is being conditioned on, because that is the slot doing the work, and the arithmetic of the ratio will follow.
Sources used in this answer
Conditionals, conditional probabilities, and conditionalization
The paper examines conditionals and conditional probabilities, focusing on their definition and usage. It clearly relates to the topic of conditional probability notation but does not address the specific convention regarding the placement of the condition.
Flipping the Conditional: Why We Are Probably Wrong About Probabilities
The paper examines errors in reasoning related to conditional probability, particularly the 'flipping the conditional' fallacy in medical and legal contexts. It defines conditional probability and provides examples, but it does not address the specific notation convention regarding the placement of the condition relative to the vertical bar.
What conditional probability could not be
The paper examines the notation and interpretation of conditional probability, specifically discussing the use of P(A|B) and the convention of placing the condition on the right side of the vertical bar. Its content is directly relevant to understanding why conditional probabilities are written with the condition on the right.
Indicative conditionals: probabilities and relevance
The paper introduces a probabilistic account of indicative conditionals, emphasizing the role of conditional probabilities and relevance. It discusses the notation p(ψ|φ) but does not address the convention regarding the placement of the condition in the notation.
Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling
The paper introduces Probability Bracket Notation and derives unified expressions for conditional expectation and conditional probability in quantum modeling. It discusses the notation for conditional probability but does not address the specific convention of placing the condition on the right side of the vertical bar.