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Conditional Probability

Conditional Probability. Knowledge about the occurrence of one event can effect the probability of occurrence for other events. The sample space is reduced to only those outcomes that are possible given the event that has occurred.

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Conditional Probability

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  1. Conditional Probability Knowledge about the occurrence of one event can effect the probability of occurrence for other events. The sample space is reduced to only those outcomes that are possible given the event that has occurred. P(A|B) - the conditional probability of A given that B has occurred. P(A|B) = P(A∩B) / P(B) where P(B)>0
  2. Conditional Probability Conditioning in probability effectively reduces the sample space. For the case of a finite sample space with equally likely outcomes, P(A|B) = N(A ∩ B) / N(B) where N(B)>0
  3. Properties of P(A|B) P(A|B) is a probability function, so has the same properties: P(A|B)≥0 P(B|B)=1 The probability of the union of A1, A2, …, Ak given that B has occurred is: P(A1 U A2 U … U Ak) = P(A1|B) + P(A2|B) + … + P(Ak|B)
  4. Multiplication Rule The probability that both A and B occur is: P(A ∩ B) = P(A|B) · P(B) or P(A ∩ B) = P(B|A) · P(A)
  5. Independence Two events are said to be independent events if the occurrence of one has no effect on the probability of the other occurring. So if A and B are independent, P(A|B) = P(A) Note: Don’t confuse independence with the concept of mutually exclusive. Two events with no outcomes in common are not independent unless one has probability 0.
  6. Independence Theorem: Events A and B are independent if and only if P(A ∩ B) = P(A) · P(B). Otherwise they are said to be dependent. If A and B are independent events, then A and B’ are independent. A’ and B are independent. A’ and B’ are independent.
  7. Redundancy The redundancy of independent components in a system can increase overall reliability of the system without having to increase the reliability of the individual components. If a system has n identical components operating identically and Fi={ith component fails}, then the probability of system failure is: P(F1∩ F2 ∩ … ∩Fn) = P(F1)n
  8. Mutual Independence Events A, B and C are mutually independent if and only if they are pairwise independent, and P(A ∩ B) = P(A) · P(B) P(A ∩ C) = P(A) · P(C) P(B ∩ C) = P(B) · P(C). P(A ∩ B ∩ C) = P(A) · P(B) · P(C)
  9. Law of Total Probability Let A1, A2, …, Am be a set of mutually exclusive and exhaustive events such that P(Ai)>0 for i=1..m. Let B be another event such that P(B)>0.
  10. Bayes Theorem Let A1, A2, …, Am be a set of mutually exclusive and exhaustive events such that P(Ai)>0 for i=1..m. Let B be another event such that P(B)>0.
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