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, then a red marble, and finally a white marble without replacement is \(\frac{1}{21}\).

Step-by-step solution

Probability of Picking a Black Marble

First, let's calculate the probability of picking a black marble from the box. There are 4 black marbles out of a total of 9 marbles. Using the formula for probability, it is: P(Black) = \(\frac{\text{Number of Black Marbles}}{\text{Total Number of Marbles}}\) P(Black) = \(\frac{4}{9}\)

Probability of Picking a Red Marble after Picking a Black Marble

Now, let's calculate the probability of picking a red marble after picking a black marble. There are 3 red marbles left in the box, and the total number of marbles has decreased to 8. Therefore, the probability is: P(Red|Black) = \(\frac{\text{Number of Red Marbles}}{\text{Total Number of Remaining Marbles}}\) P(Red|Black) = \(\frac{3}{8}\)

Probability of Picking a White Marble after Picking a Black and a Red Marble

Next, let's calculate the probability of picking a white marble after picking a black marble and then a red marble. There are 2 white marbles left in the box, and the total number of marbles has decreased again, this time to 7. Therefore, the probability is: P(White|Black, Red) = \(\frac{\text{Number of White Marbles}}{\text{Total Number of Remaining Marbles}}\) P(White|Black, Red) = \(\frac{2}{7}\)

Calculate the Overall Probability

Now we can calculate the probability of the whole sequence of events: picking a black marble first, then a red marble, and finally a white marble. To do that, we simply multiply the probabilities from steps 1, 2, and 3: P(Black, Red, White) = P(Black) × P(Red|Black) × P(White|Black, Red) P(Black, Red, White) = \(\frac{4}{9}\) × \(\frac{3}{8}\) × \(\frac{2}{7}\) P(Black, Red, White) = \(\frac{24}{504}\)

Simplify the Probability

Let's simplify the probability by dividing the numerator and the denominator by their greatest common divisor, which is 24: P(Black, Red, White) = \(\frac{24 \div 24}{504 \div 24}\) P(Black, Red, White) = \(\frac{1}{21}\) The probability of drawing a black marble, then a red marble, and finally a white marble without replacement is \(\frac{1}{21}\).