Chapter 3.6, Problem 4E

In Example 1, we wrote, “the probability that a response is yes, given that it is from a man” as $p\left(Y|M\right)$, we wrote, “the probability that a response is yes and is from a man” as $p\left(Y\cap M\right)$. In Exercise 1-4, write the given probabilities in a similar manner. Also, identify which are conditional and which are not conditional.

Let $P$ be the event that a student passes the course, and let $S$ be the event that a student studies hard. Use the symbols $P$, $S$, $|$ and $\cap$ to write the following probabilities.

a. The probability that a student passes the course given that the student studies hard.

b. The probability that a student studies hard given that thestudent passes the course

c. The probability that a student passes the course and studies hard.