# 4.5 - (4.5 Definition of conditional probability General...

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Conditional probability (4.5) Definition of conditional probability General multiplication rule Probability trees Bayes rule Independence

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Conditional probabilities reflect how the probability of an event can change if we know that some other event has occurred/is occurring. Our brains effortlessly calculate conditional probabilities, updating our “degree of belief” with each new piece of evidence .
A box contains 10 black balls and 15 white balls. Two balls are drawn randomly from the box The probability that the first ball drawn is black equals 10/25 = 0.4. What is the probability that the second ball drawn is black? Sampling with replacement The answer is obviously 10/25 = 0.4. Sampling without replacement Answer = ?

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If the first ball drawn was black, then the probability that the second ball is black would be 9/24, but if the first ball drawn was white, then the probability that the second ball is black would be 10/24. The conditional probability that the second ball is black given that the first ball was black is equal to 9/24. The conditional probability that the second ball is black given that the first ball was white is equal to 10/24. For sampling with replacement, both conditional probabilities are equal to 0.4, which is also the unconditional probability.
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