There are 3 boxes, each containing a mixture of black and white balls. Box A has 4 black balls and 6 white balls. Box B has 5 black balls and 7 white balls. Box C has 8 black balls and 2 white balls. If you pick one ball from each box without looking, what is the probability that you will have exactly two black balls?
To solve this problem, we need to consider all possible scenarios where we end up with exactly two black balls:

1. Scenario 1: We pick a black ball from Box A, a black ball from Box B, and a white ball from Box C.
- Probability of picking a black ball from Box A = 4/10
- Probability of picking a black ball from Box B = 5/12
- Probability of picking a white ball from Box C = 2/10
- Combined probability for Scenario 1 = (4/10) * (5/12) * (2/10)

2. Scenario 2: We pick a black ball from Box A, a white ball from Box B, and a black ball from Box C.
- Probability of picking a black ball from Box A = 4/10
- Probability of picking a white ball from Box B = 7/12
- Probability of picking a black ball from Box C = 8/10
- Combined probability for Scenario 2 = (4/10) * (7/12) * (8/10)

3. Scenario 3: We pick a white ball from Box A, a black ball from Box B, and a black ball from Box C.
- Probability of picking a white ball from Box A = 6/10
- Probability of picking a black ball from Box B = 5/12
- Probability of picking a black ball from Box C = 8/10
- Combined probability for Scenario 3 = (6/10) * (5/12) * (8/10)

Now, we need to add the combined probabilities of all three scenarios:
Total probability = Scenario 1 + Scenario 2 + Scenario 3