In two pots, pot1 has 5 red balls, and 7 blue balls and pot2 has 6 red balls and 12 blue balls.
3 balls are chosen from each pot.
What is the probability that all 6 are red conditional on the fact that all are the same color?
Someone rolls a die and then picks cards from a standard deck equal to the number showing on the die.
a) What is the probability that the number of jacks in the hand equals 2?
b) Conditional on knowing the number of jacks in the hand equals 2, what is the conditional probability that the die showed 3?
The Attempt at a Solution
A = probability of all same color
B = probability of all red
C = probability of all blue
all combinations for pot1 = 12x11x10 / 3x2x1 = 220
all combinations for pot1 = 18x17x16 / 3x2x1 = 816
all combinations for both pots = 220x816 = 179520
A = B u C
P(A) = P(B) + P(C) - 0
P(B) = (5choose3)(7choose0)(6choose3)(12choose0) / 179520 = 5/4488
P(C) = (5choose0)(7choose3)(6choose0)(12choose3) / 179520 = 35/816
P(A) = 5/4488 + 35/816
P(A) = 395/8976
P(all red|all same color) = P(all red ^ all same color) / P(all same color)
= (5/4488) / (395/8976) = 2/79