Conditional probability

. The unconditional probability P(A) = 0.30 + 0.10 + 0.12 = 0.52. However, the conditional probability P(AB) = 1, P(AB) = 0.12 ÷ (0.12 + 0.04) = 0.75, and P(AB) = 0. , branch probabilities are conditional on the event associated with the parent node. (Here, the overbars indicate that the event does not occur.) Conditioning on an event ==== Kolmogorov definition ==== Given two events and from the sigma-field of a probability space, with the unconditional probability of being greater than zero (i.e., , the conditional probability of given (P(A \mid B)) is the probability of A occurring if B has or is assumed to have happened. A is assumed to be the set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the quotient of the probability of the joint intersection of events and , that is, P(A \cap B), the probability at which A and B occur together, and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. For a sample space consisting of equal likelihood outcomes, the probability of the event A is understood as the fraction of the number of outcomes in A to the number of all outcomes in the sample space. Then, this equation is understood as the fraction of the set A \cap B to the set B. Note that the above equation is a definition, not just a theoretical result. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the "conditional probability of given ." As an axiom of probability Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability: :P(A \cap B) = P(A \mid B)P(B). This equation for a conditional probability, although mathematically equivalent, may be intuitively easier to understand. It can be interpreted as "the probability of B occurring multiplied by the probability of A occurring, provided that B has occurred, is equal to the probability of the A and B occurrences together, although not necessarily occurring at the same time". Additionally, this may be preferred philosophically; under major probability interpretations, such as the subjective theory, conditional probability is considered a primitive entity. Moreover, this "multiplication rule" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} As the probability of a conditional event Conditional probability can be defined as the probability of a conditional event A_B. The Goodman–Nguyen–Van Fraassen conditional event can be defined as: :A_B = \bigcup_{i \ge 1} \left( \bigcap_{j where A_i and B_i represent states or elements of A or B. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. Conditioning on an event of probability zero If P(B)=0 , then according to the definition, P(A \mid B) is undefined. The case of greatest interest is that of a random variable , conditioned on a continuous random variable resulting in a particular outcome . The event B = \{ X = x \} has probability zero and, as such, cannot be conditioned on. Instead of conditioning on being exactly , we could condition on it being closer than distance \varepsilon away from . The event B = \{ x-\varepsilon will generally have nonzero probability and hence, can be conditioned on. We can then take the limit {{NumBlk|::|\lim_{\varepsilon \to 0} P(A \mid x-\varepsilon |}} For example, if two continuous random variables and have a joint density f_{X,Y}(x,y), then by L'Hôpital's rule and Leibniz integral rule, upon differentiation with respect to \varepsilon: : \begin{aligned} \lim_{\varepsilon \to 0} P(Y \in U \mid x_0-\varepsilon The resulting limit is the conditional probability distribution of given and exists when the denominator, the probability density f_X(x_0), is strictly positive. It is tempting to define the undefined probability P(A \mid X=x) using limit (), but this cannot be done in a consistent manner. In particular, it is possible to find random variables and and values , such that the events \{X = x\} and \{W = w\} are identical but the resulting limits are not: :\lim_{\varepsilon \to 0} P(A \mid x-\varepsilon \le X \le x+\varepsilon) \neq \lim_{\varepsilon \to 0} P(A \mid w-\varepsilon \le W \le w+\varepsilon). The Borel–Kolmogorov paradox demonstrates this with a geometrical argument. Conditioning on a discrete random variable Let be a discrete random variable and its possible outcomes denoted . For example, if represents the value of a rolled dice then is the set \{ 1, 2, 3, 4, 5, 6 \}. Let us assume for the sake of presentation that is a discrete random variable, so that each value in has a nonzero probability. For a value in and an event , the conditional probability is given by P(A \mid X=x) . Writing :c(x,A) = P(A \mid X=x) for short, we see that it is a function of two variables, and . For a fixed , we can form the random variable Y = c(X, A) . It represents an outcome of P(A \mid X=x) whenever a value of is observed. The conditional probability of given can thus be treated as a random variable with outcomes in the interval [0,1]. From the law of total probability, its expected value is equal to the unconditional probability of . Partial conditional probability The partial conditional probability P(A\mid B_1 \equiv b_1, \ldots, B_m \equiv b_m) is about the probability of event A given that each of the condition events B_i has occurred to a degree b_i (degree of belief, degree of experience) that might be different from 100%. Frequentistically, partial conditional probability makes sense, if the conditions are tested in experiment repetitions of appropriate length n. Such n-bounded partial conditional probability can be defined as the conditionally expected average occurrence of event A in testbeds of length n that adhere to all of the probability specifications B_i \equiv b_i, i.e.: :P^n(A\mid B_1 \equiv b_1, \ldots, B_m \equiv b_m)= \operatorname E(\overline{A}^n\mid\overline{B}^n_1=b_1, \ldots, \overline{B}^n_m=b_m) is a special case of partial conditional probability, in which the condition events must form a partition: : P(A\mid B_1 \equiv b_1, \ldots, B_m \equiv b_m) = \sum^m_{i=1} b_i P(A\mid B_i) == Example ==

Suppose that somebody secretly rolls two fair six-sided dice, and we wish to compute the probability that the face-up value of the first one is 2, given the information that their sum is no greater than 5. • Let D1 be the value rolled on dice 1. • Let D2 be the value rolled on dice 2. '''Probability that D1 = 2''' Table 1 shows the sample space of 36 combinations of rolled values of the two dice, each of which occurs with probability 1/36, with the numbers displayed in the red and dark gray cells being D1 + D2. D1 = 2 in exactly 6 of the 36 outcomes; thus P(D1 = 2) =  = : : '''Probability that D1 + D2 ≤ 5''' Table 2 shows that D1 + D2 ≤ 5 for exactly 10 of the 36 outcomes, thus P(D1 + D2 ≤ 5) = : : '''Probability that D1 = 2 given that D1 + D2 ≤ 5 ''' Table 3 shows that for 3 of these 10 outcomes, D1 = 2. Thus, the conditional probability P(D1 = 2 | D1+D2 ≤ 5) =  = 0.3: : Here, in the earlier notation for the definition of conditional probability, the conditioning event B is that D1 + D2 ≤ 5, and the event A is D1 = 2. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference ==

In statistical inference, the conditional probability is an update of the probability of an event based on new information. The new information can be incorporated as follows: == Statistical independence ==

Events A and B are defined to be statistically independent if the probability of the intersection of A and B is equal to the product of the probabilities of A and B: :P(A \cap B) = P(A) P(B). If P(B) is not zero, then this is equivalent to the statement that :P(A\mid B) = P(A). Similarly, if P(A) is not zero, then :P(B\mid A) = P(B) is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in A and B. Independence does not refer to a disjoint event. It should also be noted that given the independent event pair [A,B] and an event C, the pair is defined to be conditionally independent if : P(AB \mid C) = P(A \mid C)P(B \mid C). This theorem is useful in applications where multiple independent events are being observed. Independent events vs. mutually exclusive events The concepts of mutually independent events and mutually exclusive events are separate and distinct. The following table contrasts results for the two cases (provided that the probability of the conditioning event is not zero). In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur). Independent vs. Positive association vs. Negative association of events == Common fallacies ==

:''These fallacies should not be confused with Robert K. Shope's 1978 "conditional fallacy", which deals with counterfactual examples that beg the question.'' Assuming conditional probability is of similar size to its inverse In general, it cannot be assumed that P(A|B) ≈ P(B|A). This can be an insidious error, even for those who are highly conversant with statistics. The relationship between P(A|B) and P(B|A) is given by Bayes' theorem: :\begin{align} P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\ \Leftrightarrow \frac{P(B\mid A)}{P(A\mid B)} &= \frac{P(B)}{P(A)} \end{align} That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B). Assuming marginal and conditional probabilities are of similar size In general, it cannot be assumed that P(A) ≈ P(A|B). These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. This fallacy may arise through selection bias. For example, in the context of a medical claim, let S be the event that a sequela (chronic disease) S occurs as a consequence of circumstance (acute condition) C. Let H be the event that an individual seeks medical help. Suppose that in most cases, C does not cause S (so that P(S) is low). Suppose also that medical attention is only sought if S has occurred due to C. From experience of patients, a doctor may therefore erroneously conclude that P(S) is high. The actual probability observed by the doctor is P(S|H). Over- or under-weighting priors Not taking prior probability into account partially or completely is called base rate neglect. The reverse, insufficient adjustment from the prior probability is conservatism. == Formal derivation ==

Formally, P(A | B) is defined as the probability of A according to a new probability function on the sample space, such that outcomes not in B have probability 0 and that it is consistent with all original probability measures. Let Ω be a discrete sample space with elementary events {ω}, and let P be the probability measure with respect to the σ-algebra of Ω. Suppose we are told that the event B ⊆ Ω has occurred. A new probability distribution (denoted by the conditional notation) is to be assigned on {ω} to reflect this. All events that are not in B will have null probability in the new distribution. For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. The former is required by the axioms of probability, and the latter stems from the fact that the new probability measure has to be the analog of P in which the probability of B is one—and every event that is not in B, therefore, has a null probability. Hence, for some scale factor α, the new distribution must satisfy: • \omega \in B : P(\omega\mid B) = \alpha P(\omega) • \omega \notin B : P(\omega\mid B) = 0 • \sum_{\omega \in \Omega} {P(\omega\mid B)} = 1. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega \notin B} P(\omega\mid B)} \\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\[5pt] &= \alpha \cdot P(B) \\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is • \omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} • \omega \notin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also ==