A bag contains 7 red and 4 black identical balls. Two balls were picked at random from the bag and replaced each time. Find the probability the two balls we...

To solve this problem, we need to determine the probability that two balls picked from the bag are of the same color. In the bag, there are 7 red balls and 4 black balls. The total number of balls in the bag is 7 + 4 = 11.

We will approach this by calculating the probabilities of two events: both balls being red, and both balls being black. Then, we'll add these probabilities together to get the total probability that the balls are the same color.

Step 1: Probability both balls are red.

The probability of picking a red ball first is 7/11. After replacing the ball, the probability of picking a red ball again is still 7/11 because the condition remains the same. Therefore, the probability of picking two red balls is:

(7/11) * (7/11) = 49/121

Step 2: Probability both balls are black.

The probability of picking a black ball first is 4/11. After replacing the ball, the probability of picking a black ball again is still 4/11. Thus, the probability of picking two black balls is:

(4/11) * (4/11) = 16/121

Step 3: Total probability of both balls being the same color.

To find the total probability that the two balls are of the same color (either both red or both black), we add the probabilities from Step 1 and Step 2:

(49/121) + (16/121) = 65/121

Therefore, the probability that the two balls picked are of the same color is 65/121.