Conditional Probability And Independence

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What is Conditional Probability?

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

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What is the formula for Conditional Probability?

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P(A|B) = P(A and B) / P(B), provided P(B) > 0.

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What does it mean for two events to be 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, P(A|B) = P(A) and P(B|A) = P(B).

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How do you calculate the probability of two independent events both occurring?

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For independent events A and B, the probability of both occurring is P(A and B) = P(A) * P(B).

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What is the Law of Total Probability?

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If B1, B2, ..., Bn are mutually exclusive events that cover the entire sample space, then P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn).

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What is the Complement Rule in probability?

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

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How do you find P(A and B) for dependent events?

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For dependent events, P(A and B) = P(A) * P(B|A).

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What is Bayes' Theorem?

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

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What is the difference between joint probability and conditional probability?

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Joint probability P(A and B) is the probability of both events occurring together, while conditional probability P(A|B) is the probability of A occurring given that B has occurred.

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What is a probability distribution?

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A probability distribution describes how the probabilities are distributed over the values of a random variable.