A bag contains 5 red balls and 7 black balls. Two balls are drawn at random one after another without replacement. Find the probability that both drawn balls are red.

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Step-by-Step Solution

Step 1: Identify total number of balls and red balls

First, we identify the total number of red balls and black balls in the bag. Then, we calculate the total number of balls in the bag by adding the number of red balls and black balls.

Step 2: Calculate probability of drawing the first red ball

The probability of drawing the first red ball is the ratio of the number of red balls to the total number of balls in the bag.

Step 3: Calculate probability of drawing the second red ball (without replacement)

Since the first ball drawn was red and it was not replaced, the number of red balls decreases by 1, and the total number of balls in the bag also decreases by 1. We then calculate the probability of drawing another red ball from the remaining balls.

Step 4: Calculate the probability of both events occurring

To find the probability that both drawn balls are red, we multiply the probability of drawing the first red ball by the conditional probability of drawing the second red ball given that the first was red. Finally, we simplify the fraction.