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Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago? See more

Conditional Probability Misconceptions

         

Which of the following equalities is not necessarily true if \(A, B\) are events?

Suppose you flip a fair coin 10 times. What is the probability of the last two flips both being heads if you know that the first eight flips were heads?

\[ P(A \mid B) = P(B \mid A).\]

which of the following must be true of \( A \) and \( B, \) if \(P(A \cap B) > 0?\)

Suppose there are two types of surgeries that doctors perform. Doctor A has a higher success rate than Doctor B on the first type of surgery. Doctor A also has a higher success rate than Doctor B on the second type of surgery. Is it true that Doctor A necessarily has a higher overall success rate than Doctor B?

Assume that a child has an equal chance of being born on any month and of either gender (male, female). Let \( p \) be the probability that given that a family has two children and at least one of them is a girl, both are girls. Furthermore, let \( q \) be the probability that given that a family has two children and at least one of them is a girl born in April, that both are girls. Does \( p = q \)?

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