Probability

# 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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