Question

A bag contains 4 black and 5 blue marbles. A marble is drawn and then replaced, after which a second marble is drawn. What is the probability that the first is black and second blue?

Short answer

The probability that the first marble is black and the second marble is blue after the first marble has been replaced is \(P(Black \ and \ Blue) = \frac{20}{81}\).

Step-by-step solution

Find the probability of drawing a black marble.

First, we need to find the probability of drawing a black marble from 4 black marbles and 5 blue marbles in the bag. The total number of marbles in the bag is 4 + 5 = 9. The probability of drawing a black marble is the ratio of the number of black marbles to the total number of marbles, which is P(Black) = \(\frac{4}{9}\).

Find the probability of drawing a blue marble.

Next, we need to find the probability of drawing a blue marble from the bag after replacing the first marble. Since the first marble is replaced, the total number of marbles in the bag remains 9. The probability of drawing a blue marble is the ratio of the number of blue marbles to the total number of marbles, which is P(Blue) = \(\frac{5}{9}\).

Apply the rule for the probability of independent events.

Now, we need to find the probability that the first marble is black and the second marble is blue. Since the marbles are replaced after each draw, the two events are independent. The probability of two independent events occurring is the product of their individual probabilities. Therefore, the probability of drawing a black marble followed by a blue marble is P(Black and Blue) = P(Black) × P(Blue) = \(\frac{4}{9}\) × \(\frac{5}{9}\).

Calculate the required probability.

After multiplying the probabilities from Step 3, we have P(Black and Blue) = \(\frac{20}{81}\). This is the probability that the first marble is black and the second marble is blue after the first marble has been replaced.