Question

A box contains 4 black marbles, 3 red marbles, and 2 white marbles. What is the probability that a black marble, then a red marble, then a white marble is drawn without replacement?

Short answer

The probability of drawing a black marble, followed by a red marble, and finally a white marble without replacement is \(\frac{1}{21}\).

Step-by-step solution

Determine the probability of drawing a black marble

First, we need to find the probability of drawing a black marble. There are 4 black marbles and a total of 9 marbles (4 black, 3 red, and 2 white). Thus, the probability of drawing a black marble is the ratio of black marbles to the total number of marbles, which is: \(P(B) = \frac{4}{9}\)

Determine the probability of drawing a red marble given that a black marble was drawn first

After drawing a black marble, there are now 8 marbles left in the box (3 black, 3 red, and 2 white). We need to find the probability of drawing a red marble given that a black marble was previously removed. Since there are 3 red marbles out of 8 remaining marbles, the probability is: \(P(R|B) = \frac{3}{8}\)

Determine the probability of drawing a white marble given that a black and a red marble were drawn before

Now, there are 7 marbles left in the box (3 black, 2 red, and 2 white). After drawing a black marble and a red marble, we need to find the probability of drawing a white marble. There are 2 white marbles out of 7 remaining marbles, so the probability is: \(P(W|R,B) = \frac{2}{7}\)

Calculate the overall probability

We found the probability of each event separately, but we need to find the probability of all three events happening in sequence (black marble first, red marble second, and white marble third). To do this, we need to multiply the probabilities of each event as they are independent events: \(P(B, R, W) = P(B) * P(R|B) * P(W|R,B)\) Plug in the values from previous steps: \(P(B, R, W) = \frac{4}{9} * \frac{3}{8} * \frac{2}{7}\)

Simplify the expression

Multiply the fractions together to get the final probability: \(P(B, R, W) = \frac{24}{504}\) Simplify the fraction: \(P(B, R, W) = \frac{1}{21}\) Therefore, the probability of drawing a black marble, followed by a red marble, and finally a white marble without replacement is \(\frac{1}{21}\).