Conditional Probability

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What is Conditional Probability?

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Conditional Probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as 'the probability of A given B'.

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How is Conditional Probability calculated?

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Conditional Probability is calculated using the formula: P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability of both events A and B occurring.

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What does it mean if two events are independent?

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Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, this is expressed as P(A|B) = P(A) or P(B|A) = P(B).

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What is the formula for the probability of the union of two events?

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The probability of the union of two events A and B is given by: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

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What is the complement of an event?

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The complement of an event A, denoted as A', is the event that A does not occur. The probability of the complement is given by P(A') = 1 - P(A).

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How do you find the probability of an event not occurring?

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The probability of an event A not occurring is found using the formula: P(A') = 1 - P(A).

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What is Bayes' Theorem?

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Bayes' Theorem relates the conditional and marginal probabilities of random events. It is expressed as: P(A|B) = [P(B|A) * P(A)] / P(B).

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What is the Law of Total Probability?

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The Law of Total Probability states that if B1, B2, ..., Bn are disjoint events that cover the entire sample space, then for any event A: P(A) = Σ P(A|Bi) * P(Bi).

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What is the difference between P(A|B) and P(B|A)?

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P(A|B) is the probability of A occurring given that B has occurred, while P(B|A) is the probability of B occurring given that A has occurred. They are not generally equal.

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What is a joint probability?

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Joint probability is the probability of two events A and B occurring together, denoted as P(A ∩ B).