I'm studying probability. This is not homework. I have been studying for a graduate master's since September 2015. The textbook is Probability : An Introduction (Grimmett & Welsh).
You are presented with two urns. Urn 1 contains 3 white and 4 black balls, and Urn 2 contains 2 white and 6 black balls. (a) You pick a ball randomly from Urn 1 and place it in Urn II. Next you pick a ball randomly from Urn 2. What is the probability that the ball is black?
I made the following reasoning.
Prob of picking black from the Urn 1 = P(B1) = 4/7
Prob of picking white from the Urn 1 = P(W1) = 3/7
Prob of picking black from the Urn 2, after placing the picked ball in it = P(B2) = ?
B1 and W1 are a partition of Omega so I can use the partition theorem:
P(B2) = P(B2|B1)*P(B1) + P(B2|W1)*P(W1)
I intuitively know that P(B2|B1) = 7/9, because, given that I picked a black ball from the Urn 1 and placed it in the Urn II, now I have 7 possibility of success and 9 possible outcomes.
I intuitively know that P(B2|W1) = 6/9 for the same reason.
Then P(B2) = 7/9*4/7+6/9*37 = 0.73
I would like to know if my reasoning is correct and what is the formal way to show that P(B2|B1) = 7/9 and P(B2|W1) = 6/9.