A bag contains 3 red balls 5 black balls . Two balls are to be  taken at  random without replacement, from the bag. Find the probability that the two balls taken  are of the same colour.

Sure, let's solve this step by step:

Step 1: Identify the total number of balls in the bag. We have 3 red balls and 5 black balls, so the total number of balls is 3 + 5 = 8.

Step 2: Calculate the probability of drawing two balls of the same color. This can happen in two ways: either both balls are red or both balls are black.

Step 3: Calculate the probability of drawing two red balls. The probability of drawing the first red ball is 3/8. Since we are not replacing the balls, the probability of drawing a second red ball is 2/7. So, the probability of drawing two red balls is (3/8) * (2/7) = 6/56 = 3/28.

Step 4: Calculate the probability of drawing two black balls. The probability of drawing the first black ball is 5/8. The probability of drawing a second black ball is 4/7. So, the probability of drawing two black balls is (5/8) * (4/7) = 20/56 = 10/28 = 5/14.

Step 5: Add the two probabilities together. The probability of drawing two balls of the same color is 3/28 + 5/14 = 6/28 + 10/28 = 16/28 = 4/7.

So, the probability that the two balls drawn are of the same color is 4/7.