A big box contains 3 black balls and 7 white balls. In Step 1, half of the balls are taken out of the big box (without replacement) and put in a small box, which was originally empty.
(a) For each x = 1, 2, 3 Find the probabilities that x black balls and 5-x white balls were transferred
My work:
P(x) = ((3 choose x)(7 choose 5-x))/(10 choose 5)
P(x=1) = 0.417
P(x=2) = 0.417
P(x=3) = 0.083
(b) In Step 2, one ball is then drawn at random from the small box. Find the probability that this ball is black, if x black balls and 5-x white balls were transferred, for each x = 1, 2, 3.
My work:
x=1 means there are 4 white balls and 1 black ball so the probability of the ball being black when pulled is 1/5 = 0.2
x=2 means there are 3 white balls and 2 black balls so the probability of the ball being black when pulled is 2/5 = 0.4
x=3 means there are 2 white balls and 3 black balls so the probability of the ball being black when pulled is 3/5 = 0.6
(c) In part (b) suppose the randomly drawn ball is observed to be black. What is the probability that 2 black balls and 3 white balls were transferred from the big box to the small box?
My work:
I was thinking maybe I should consider the different combinations of 2 black and 3 white so I did this:
BBWWW
BWBWW
BWWBW
BWWWB
WBWWB
WWBWB
WWWBB
WWBBW
WBBWW
WBWBW
But after that I am lost.
Are these anywhere near correct? I would really appreciate any feedback and help that I can get on these problems. Thank you!
(a) For each x = 1, 2, 3 Find the probabilities that x black balls and 5-x white balls were transferred
My work:
P(x) = ((3 choose x)(7 choose 5-x))/(10 choose 5)
P(x=1) = 0.417
P(x=2) = 0.417
P(x=3) = 0.083
(b) In Step 2, one ball is then drawn at random from the small box. Find the probability that this ball is black, if x black balls and 5-x white balls were transferred, for each x = 1, 2, 3.
My work:
x=1 means there are 4 white balls and 1 black ball so the probability of the ball being black when pulled is 1/5 = 0.2
x=2 means there are 3 white balls and 2 black balls so the probability of the ball being black when pulled is 2/5 = 0.4
x=3 means there are 2 white balls and 3 black balls so the probability of the ball being black when pulled is 3/5 = 0.6
(c) In part (b) suppose the randomly drawn ball is observed to be black. What is the probability that 2 black balls and 3 white balls were transferred from the big box to the small box?
My work:
I was thinking maybe I should consider the different combinations of 2 black and 3 white so I did this:
BBWWW
BWBWW
BWWBW
BWWWB
WBWWB
WWBWB
WWWBB
WWBBW
WBBWW
WBWBW
But after that I am lost.
Are these anywhere near correct? I would really appreciate any feedback and help that I can get on these problems. Thank you!