Conditional Probability Question

[randomcat]

Suppose that there are 2 apple trees. Tree A and tree B.

A produces 0.7 of the farm's apples. And B produces 0.3.

Out of the apples that tree A produces, 0.15 are bad. For B, 0.05 are bad.

One package of goodies contains 3 apples.

Given this information, what is the P(Tree A| at least one bad apple in the package)

Okay, so here's what I tried.

Probability of at least 1 bad apple in the package from tree A is:
x=0.7*(1-(0.85)^3)
Probability of at least 1 bad apple in the package from tree B is:
y=0.3*(1-(0.95)^3)

Then P(Tree A| at least one bad apple in the package) = x/(x+y) = 0.863

Does this seem right? Any help would be greatly appreciated.

 

[mathman]

An exact, but kludgy approach would be using a multinomial (4) distribution.
A = prob from tree A and good = .595 (no. = j)
a = prob from tree A and bad = .105 (no. = k)
B = prob from tree B and good = .285 (no. = m)
b = prob from tree B and bad = .015 (no. = n)

term = {3!/(j!k!m!n!)}A^ja^kB^mbⁿ where j,k,m,n ≥ 0 and j+k+m+n = 3

Desired probability is fraction.
Numerator = sum of terms with k > 0 (contains a bad apple from A).
Denominator = sum of terms with k > 0 or n > 0 (contains a bad apple).