Probability of Drawing a Black Marble from Two Boxes

Consider two boxes, each containing a different number of black and white marbles. One box contains one black and one white marble, while the other box contains two black and one white marble. If we select a box at random and then draw a marble from the selected box, what is the probability that the marble will be black?

To answer this question, we need to first find the probability of selecting each box and then the probability of drawing a black marble from each box. Let's assume that the probability of selecting each box is equal, so both boxes have a probability of 0.5 of being selected.

If we select the first box, the probability of drawing a black marble is 1/2. If we select the second box, the probability of drawing a black marble is 2/3. To find the overall probability of drawing a black marble, we need to take into account the probability of selecting each box and the probability of drawing a black marble from each box.

The overall probability of drawing a black marble is calculated as follows:

P(black) = P(box 1 selected) * P(black from box 1) + P(box 2 selected) * P(black from box 2)

P(black) = 0.5 * 1/2 + 0.5 * 2/3

P(black) = 0.5 + 1/3

P(black) = 5/6

So, the probability of drawing a black marble from two boxes, one containing one black and one white marble and the other containing two black and one white marble, is 5/6.

In summary, understanding the probability of drawing a black marble from two boxes requires us to consider the probability of selecting each box and the probability of drawing a black marble from each box. By combining these probabilities, we can find the overall probability of drawing a black marble, which in this case is 5/6.