Conditional probability

In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, 'the conditional probability of A given B', or 'the probability of A under the condition B', is usually written as P(A | B), or sometimes PB(A) or P(A / B). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%; P(Cough | Sick) = 75% In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, 'the conditional probability of A given B', or 'the probability of A under the condition B', is usually written as P(A | B), or sometimes PB(A) or P(A / B). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%; P(Cough | Sick) = 75% The concept of conditional probability is one of the most fundamental and one of the most important in probability theory. But conditional probabilities can be quite slippery and require careful interpretation. For example, there need not be a causal relationship between A and B, and they don't have to occur simultaneously. P(A | B) may or may not be equal to P(A) (the unconditional probability of A). If P(A | B) = P(A), then events A and B are said to be 'independent': in such a case, knowledge about either event does not give information on the other. P(A | B) (the conditional probability of A given B) typically differs from P(B | A). For example, if a person has dengue, they might have a 90% chance of testing positive for dengue. In this case what is being measured is that if event B ('having dengue') has occurred, the probability of A (test is positive) given that B (having dengue) occurred is 90%: that is, P(A | B) = 90%. Alternatively, if a person tests positive for dengue they may have only a 15% chance of actually having this rare disease because the false positive rate for the test may be high. In this case what is being measured is the probability of the event B (having dengue) given that the event A (test is positive) has occurred: P(B | A) = 15%. Falsely equating the two probabilities causes various errors of reasoning such as the base rate fallacy. Conditional probabilities can be reversed using Bayes' theorem. Conditional probabilities can be displayed in a conditional probability table. Given two events A and B, from the sigma-field of a probability space, with the unconditional probability of B (that is, of the event B occurring ) being greater than zero, P(B) > 0, the conditional probability of A given B is defined as the quotient of the probability of the joint of events A and B, and the probability of B: where P ( A ∩ B ) {displaystyle P(Acap B)} is the probability that both events A and B occur. This may be visualized as restricting the sample space to situations in which B occurs. The logic behind this equation is that if the possible outcomes for A and B are restricted to those in which B occurs, this set serves as the new sample space. Note that this is a definition but not a theoretical result. We just denote the quantity P ( A ∩ B ) P ( B ) {displaystyle {frac {P(Acap B)}{P(B)}}} as P ( A ∣ B ) {displaystyle P(Amid B)} and call it the conditional probability of A given B. Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability: Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations such as the subjective theory, conditional probability is considered a primitive entity. Further, this 'multiplication axiom' introduces a symmetry with the summation axiom for mutually exclusive events: