Conditional Probability

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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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How is Conditional Probability calculated?

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P(A | B) = P(A and B) / P(B), where P(A and B) is the probability of both events occurring and P(B) is the probability of event B.

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What does P(A | B) represent?

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It represents the probability of event A occurring given that event B has occurred.

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If P(A) = 0.5 and P(B) = 0.4, what is P(A | B) if P(A and B) = 0.2?

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P(A | B) = P(A and B) / P(B) = 0.2 / 0.4 = 0.5.

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What is the formula for the Total Probability Theorem?

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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 Law of Total Probability?

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It states that the total probability of an event can be found by considering all the different ways that event can occur, weighted by the probabilities of those ways.

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What is the difference between independent and dependent events?

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Independent events do not affect each other's probabilities (P(A | B) = P(A)). Dependent events do affect each other's probabilities (P(A | B) ≠ P(A)).

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What is an example of Conditional Probability in real life?

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Determining the probability of a person being a smoker given that they have lung cancer.

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What is the probability of drawing a red card from a standard deck of cards?

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P(Red) = 26/52 = 1/2.

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What is the probability of drawing a heart given that a card drawn is a red card?

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P(Heart | Red) = 13/26 = 1/2.