Conditional Probability

Front

What is Conditional Probability?

Back

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

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What is the formula for Conditional Probability?

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

Front

What does P(A|B) represent?

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P(A|B) represents the probability of event A occurring given that event B has occurred.

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How do you interpret P(B|A)?

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P(B|A) represents the probability of event B occurring given that event A has occurred.

Front

What is the relationship between independent events and Conditional Probability?

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For independent events, P(A|B) = P(A). This means the occurrence of event B does not affect the probability of event A.

Front

If P(A) = 0.5 and P(B) = 0.4, what is P(A ∩ B) if A and B are independent?

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If A and B are independent, P(A ∩ B) = P(A) * P(B) = 0.5 * 0.4 = 0.2.

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What is the Total Probability Theorem?

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The Total Probability Theorem states that 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 Bayes' Theorem?

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

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How do you calculate the probability of a student playing a sport given they play an instrument?

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Use the formula for conditional probability: P(Sport|Instrument) = P(Sport ∩ Instrument) / P(Instrument).

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What is the significance of the intersection in Conditional Probability?

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The intersection P(A ∩ B) represents the probability that both events A and B occur simultaneously.