Probability of Selecting the First Box Given a White Marble is Drawn

Consider two boxes, one containing one black and one white marble, and the other containing two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the first box was selected, given that the marble drawn is white?

To solve this problem, we can use Bayes' theorem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B. In this case, event A is the selection of the first box and event B is the drawing of a white marble.

First, let's find the probability of drawing a white marble, P(B). We have:

P(B) = P(first box selected) * P(white from first box) + P(second box selected) * P(white from second box)

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

P(B) = 0.5 + 1/6

P(B) = 2/3

Next, let's find the probability of the first box being selected given a white marble is drawn, P(A | B). Using Bayes' theorem, we have:

P(A | B) = P(B | A) * P(A) / P(B)

P(B | A) is the probability of drawing a white marble given the first box is selected. This is equal to 1/2. P(A) is the probability of selecting the first box, which is 0.5.

P(A | B) = 1/2 * 0.5 / (2/3)

P(A | B) = 1/4

So, the probability that the first box was selected given that a white marble was drawn is 1/4.

In conclusion, to find the probability of selecting the first box given a white marble is drawn, we can use Bayes' theorem. By calculating the probability of drawing a white marble and the probability of drawing a white marble given the first box is selected, we can find the overall probability of the first box being selected. In this case, the probability of selecting the first box given a white marble was drawn is 1/4.