Probability?

Question

One urn contains 7 white balls and 3 black balls, and a second urn contains 4 white balls and 2 black balls. A ball is selected from each urn, and is placed in a bag containing 5 white and 6 black balls. What is the probability of drawing of drawing a white ball from the bag?

Answer 1

Use decision tree.
These are the possible cases:
no W ball from both urn: probability = 3/10 * 2/6 = 1/10=3/30
only 1 W ball from 1st urn: probability = 7/10 * 2/6 = 7/30
only 1 W ball from 2nd urn: probability = 3/10 * 4/6 = 1/5=6/30
2 W balls from both urns: probability = 7/10 * 4/6 = 14/30
(Check, sum of probabilities = 30/30 = 1)

probability of a W ball from the bag for the above 4 cases are:
5/13, 6/13, 6/13, 7/13

Thus the probability of drawing a white ball from the bag is
3/30*5/13 + 7/30*6/13 + 6/30*6/13 + 14/30*7/13
= (15+42+36+98)/390
= 191/390

Answer 2

First calculate the probability of drawing zero, one, or two white balls from the first two urns.

P(0) = (3/10)(2/6) = 1/10
P(2) = (7/10)(4/6) = 7/15
P(1) = 1 - P(0) - P(2) = 1 - 1/10 - 7/15 = 13/30

This give us the probabilities for the three possible compositions of the third urn. After adding the two drawn balls from the first two urns to the third urn we have:

P(5W & 8B) = 1/10
P(6W & 7B) = 13/30
P(7W & 6B) = 7/15

The probability of drawing a white ball from the third urn is:

P(W) = (1/10)(5/13) + (13/30)(6/13) + (7/15)(7/13)
P(W) = 191/390

Answer 3

suppose (u_1,u_2) represents urn 1 and urn 2. you make selections as follows....
let w=white and b=black
(w,w),(b,b),(w,b),(b,w)..
after you make the selections you know what the probability of selecting a white ball from the bag will be. You also know what the probability of selecting the previous combinations are...
so, let A = a white ball is select from the bag.
P(A)= P(A|W intersect W)P(W intersect W) + P(A| B intersect B)P(B intersect B) + P(A|W intersect B)P(W intersect B) + P(A|B intersect W)P(B intersect W)....

About your other stat question..
I am 90% sure the mean is 30 and the probability in question is P(X=>2). That questions rang a bell :P