Law Of Total Probability Law Of Total Probability The law of total probability states that if (An : n = 1, 2, 3…) is a finite part of a sample space and event An is a measurable then for the other event B of the same probability space. P ( B ) = ∑n P (B∩ An ) P ( B ) = ∑n P ( B |An) Pr ( An) Here for any n for which P (An) = 0, these terms can be simply omitted from the summation part because P ( B |An) is finite. The term law of probability is sometimes also known as law of alternatives. The law of total probability theorem is often written as follows: P( A ) = P ( A ∩ B ) + P ( A ∩ B’ ), Where P ( A ) is the probability that the event A occurs. P ( A ∩ B ) refers that the event A and B both occur and P ( A ∩ B’ ) is that event A and B’ both occur means A occurs but B doesn’t. Using the multiple rules the following expression can be written: P ( A ) = P ( A | B ). P ( B ) + P ( A |B’ ). P ( B’ ) Baye’s theorem is related to the concept of law of probability. Baye’s theorem Know More About Real Number Worksheet
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provides a new rule with the use of multiplication rule that can be written as follows: P ( A | B ) = P ( A ∩ B ) / P ( B ) = P ( B | A ) P ( A ) / P ( B ), This is the simplest form of Baye’s theorem. Using the law of total probability Baye’s theorem can be written as: P ( A | B ) = P ( B | A ) P ( A ) / P ( B | A ). P ( A ) + P ( B | A’ )P ( A’ ) Where, P ( A ) is the probability that the event A occurs only. P ( B ) is the probability that the event B occurs only. P ( A’ ) is the event that A doesn’t occur. P ( A | B ) is the probability given that event A occurs that event B has already occurred. P ( B | A ) is the probability that event B occurs given that event A has already occurred. P ( B | A’ ) is the probability that event B occurs given that event A has already not occurred. In short the law of total probability theorem can be written as: if B1, B2, … Bn is a partition of S such that p ( Bi ) > 0 for i = 1….n then for any event A. P ( A ) = ∑i=1….n P ( A | Bi ) p ( Bi ). The law of total probability is the proposition that if \left\\right\} is a finite or countably infinitepartition of a sample space (in other words, a set of pairwise disjointevents whose union is the entire sample space) and each event B_n is measurable, then for any event A of the same probability space: Read More About Rational Word Problems
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\Pr(A)=\sum_{n} \Pr(A\cap B_n)\, or, alternatively, \Pr(A)=\sum_{n} \Pr(A\mid B_n)\Pr(B_n)\,, where, for any n\, for which \Pr(B_n) = 0 \, and hence \Pr(A\mid B_n)\, is not defined, we define "undefined times zero" equals zero. Equivalently, these n\, are simply omitted from the summation. One common application of the law is where the events coincide with a discrete random variableX taking each value in its range, i.e. B_n is the event X=x_n. It follows that the probability of an event A is equal to the expected value of the conditional probabilities of A given X=x_n. That is, \Pr(A)=\sum_{n} \Pr(A\mid X=x_n)\Pr(X=x_n) = \operatorname{E}_X[\Pr(A\mid X)] , where Pr(A|X) is the conditional probability of A given X, and where EX denotes the expectation with respect to the random variable X. This result can be generalized to continuous random variables, and the expression becomes \Pr(A)= \operatorname{E}[\Pr(A\mid \mathcal{F}_{X})], where \mathcal{F}_{X} denotes the sigma-algebra generated by the random variable X.
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