Conditional Probability
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 \( p = q \)?