Problem:
A bag contains 6 white and 4 black balls. Two balls are drawn at random. Find the probability that they are of the same color.

Solution:

Step 1: Understanding the problem
We are given a bag that contains 6 white and 4 black balls. We need to find the probability that when two balls are drawn at random, they will be of the same color.

Step 2: Identifying the total number of outcomes
To find the total number of outcomes, we need to calculate the number of ways two balls can be drawn from a bag containing 10 balls. This can be calculated using the combination formula:

nCr = n! / ((n-r)! * r!)

Where n is the total number of balls and r is the number of balls drawn.

In this case, n = 10 and r = 2, so the total number of outcomes is:

10C2 = 10! / ((10-2)! * 2!) = 45

Step 3: Identifying the favorable outcomes
To find the favorable outcomes, we need to calculate the number of ways two balls of the same color can be drawn.

Case 1: Drawing two white balls
The number of ways to draw two white balls from a bag containing 6 white balls can be calculated using the combination formula:

6C2 = 6! / ((6-2)! * 2!) = 15

Case 2: Drawing two black balls
Similarly, the number of ways to draw two black balls from a bag containing 4 black balls can be calculated using the combination formula:

4C2 = 4! / ((4-2)! * 2!) = 6

Therefore, the total number of favorable outcomes is 15 + 6 = 21.

Step 4: Calculating the probability
The probability that two balls drawn at random are of the same color can be calculated using the formula:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

In this case, the probability is:

Probability = 21 / 45 = 7/15

Therefore, the probability that two balls drawn at random are of the same color is 7/15.

Conclusion:
When two balls are drawn at random from a bag containing 6 white and 4 black balls, the probability that they are of the same color is 7/15.