Conditional Probability And Independence

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What is Conditional Probability?

Back

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, P(A and B) = P(A) * P(B).

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What is 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 theoretical probability and experimental probability?

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Theoretical probability is based on the possible outcomes in a perfect world, while experimental probability is based on actual experiments and observed outcomes.

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How do you find the probability of drawing a specific color marble from a bag?

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Divide the number of favorable outcomes (specific color marbles) by the total number of outcomes (total marbles in the bag).

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What is the probability of drawing two marbles of the same color without replacement?

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P(same color) = (number of first color marbles / total marbles) * (number of remaining same color marbles / remaining total marbles).

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What is the probability of flipping a coin and then spinning a spinner?

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Multiply the probability of each independent event: P(Heads) * P(Yellow). For a fair coin and a spinner with equal sections, P(Heads) = 1/2 and P(Yellow) = 1/n (where n is the number of sections).