From a group of 20 students, 8 of whom are computer science majors and 12 of whom are electrical engineering majors, one student is selected at random, and then a second student is selected at random from the remaining 19 students. Let A be the event that the first student is a computer science major,and let B be the event that the second student is a computer science major. Prove that A and B are dependent events.
I tried to use P(A|B)=P(A intersect B)/P(B) should not be equal to P(A), but I'm running into trouble with the numbers. Is P(A intersect B)=(8/20)(7/19)? And is P(B)=(8/19) or (7/19)? Very confused.
Best Answer
Yes, $P(A\cap B)$ is correct. You need Bayes's formula to do $P(B)$: $$P(B) = P(B|A)P(A)+ P(B|\text{not }A)P(\text{not }A) = \frac7{19}\cdot\frac8{20} + \frac8{19}\cdot\frac{12}{20} = \frac25.$$ Note that $P(B|A)P(A) = P(A\cap B) = P(A|B)P(B)$.