Question

There are 6 white, 5 black and 10 red balls in the box. We take out three balls from the box, what is the probability that the balls taken out are different colors?

Answer 1

The question involves calculating the probability of drawing three balls of different colors from a box with a mix of white, black, and red balls. The answer requires combination calculations to find the number of favorable and total outcomes. The probability is then determined as the ratio of these two numbers.

Step-by-step explanation:

The student is asking about the probability of drawing three balls of different colors from a box containing balls of three distinct colors. We have 6 white, 5 black, and 10 red balls, and want to calculate the probability that one ball of each color is drawn when taking out three balls.

Step-by-step Solution

1. Calculate the total number of ways to draw three balls from the box, regardless of color.
2. Calculate all the possible outcomes for drawing one ball of each color.
3. Compute the probability by dividing the number of favorable outcomes by the total number of outcomes.

To find the total number of ways to draw three balls (denominator), we combine the counts of each color to get a total of 21 balls. The number of ways to select 3 balls out of 21 is given by the combination formula C(n, k) which is n! / (k!(n-k)!). So we have C(21, 3).

For the favorable outcomes (numerator), we draw one ball of each color. This can be done in C(6, 1) * C(5, 1) * C(10, 1) ways.

Finally, we calculate the probability by dividing the number of favorable outcomes by the total number, obtaining probability = [C(6, 1) * C(5, 1) * C(10, 1)] / C(21, 3).

Answer 2

The probability that the balls taken out are different colors is 0.464

We solve this using complementary probability

Calculating the probability of NOT choosing 3 different colors:

Probability of choosing 3 white balls: (6/21) * (5/20) * (4/19) = 0.1176

Probability of choosing 3 black balls: (5/21) * (4/20) * (3/19) = 0.0758

Probability of choosing 3 red balls: (10/21) * (9/20) * (8/19) = 0.3423

Total probability of not choosing different colors: 0.1176 + 0.0758 + 0.3423 = 0.5357

Lets subtract the complementary probability from 1 to get the probability of choosing 3 different colors:

= 1 - 0.5357 ≈ 0.464

Therefore, the probability that the balls taken out are different colors is 0.464