The Question: Suppose you have a class with 30 students ‒ 10 freshmen, 12 sophomores, and 8 juniors.
You pick two students at random, one at a time.
What is the probability that the second student is a freshman, given that the first is a freshman?
I know how to solve it in a different way, but I don't know how to solve it with Conditional Probability. So I need help solving it with Conditional Probability.
Thank you in advance.
Best Answer
The probability of "the second student is a freshman" given "the first student is a freshman" is the probability of both occurring simultaneously divided by the probability of "the first student is a freshman". That is, assigning $A = {}$ "the second student is a freshman" and $B = {}$ "the first student is a freshman", and using $P$ to represent probability, $$ P(A \mid B) = \frac{P(A \cap B)}{P(B)} $$
Now $P(B) = 10/30 = 1/3$, is easy enough to get from the given data. One way to compute $P(A \cap B)$ is $$ P(A \cap B) = \frac{10}{30} \cdot \frac{9}{29} = \frac{9}{3 \cdot 29} \text{.} $$ (This is not the only way to get this probability.) Then $$ P(A|B) = \frac{\frac{9}{3 \cdot 29}}{\frac{1}{3}} = \frac{9}{29} \text{.} $$