Suppose that a certain college class contains 35 students. Of these, 22 are seniors, 19 are physics majors, and 12 are neither. A student is selecte from the class.
(a) What is the probability that the student is both a senior and a physics major?
(b) Given that the student selected is a senior, what is the probability that she is also a physics major?

Write your responses as fractions. (If necessary, consult a list of formulas.)
(a) [
(b) $\square$

Step 1 ：Let's denote the total number of students as \(N\), the number of seniors as \(S\), the number of physics majors as \(P\), and the number of students who are neither seniors nor physics majors as \(N\).

Step 2 ：We are given that \(N = 35\), \(S = 22\), \(P = 19\), and \(N = 12\).

Step 3 ：We are asked to find the probability that a randomly selected student is both a senior and a physics major. This is equivalent to finding the intersection of the set of seniors and the set of physics majors.

Step 4 ：We can find this intersection by adding the number of seniors and physics majors and subtracting the total number of students and those who are neither. This gives us \(S + P - N - N = 22 + 19 - 35 - 12 = 18\).

Step 5 ：The probability is then the ratio of this intersection to the total number of students, which is \(\frac{18}{35}\) or approximately 0.514.

Step 6 ：\(\boxed{\text{The probability that the student is both a senior and a physics major is } \frac{18}{35} \text{ or approximately 0.514.}}\)

Step 7 ：We are also asked to find the conditional probability that a student is a physics major given that they are a senior. This is the ratio of the intersection of seniors and physics majors to the total number of seniors, which is \(\frac{18}{22}\) or approximately 0.818.

Step 8 ：\(\boxed{\text{Given that the student selected is a senior, the probability that she is also a physics major is } \frac{18}{22} \text{ or approximately 0.818.}}\)