Two balls are taken at the same time from a bag containing three black, three yellow and three red balls. (i) Find the probability that both balls are red. (i)

To find the probability that both balls drawn are red, we'll follow these steps: ### **Given:** - Total balls in the bag: - Black balls = 3 - Yellow balls = 3 - Red balls = 3 - Total number of balls = 3 + 3 + 3 = **9** ### **Step 1: Calculate the Total Number of Ways to Choose 2 Balls** The total number of ways to choose 2 balls out of 9 is given by the combination formula: \[ \text{Total combinations} = C(9, 2) = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \] ### **Step 2: Calculate the Number of Favorable Ways (Both Balls are Red)** There are 3 red balls. The number of ways to choose 2 red balls out of these 3 is: \[ \text{Red combinations} = C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \] ### **Step 3: Calculate the Probability** The probability \( P \) that both balls drawn are red is the ratio of the favorable combinations to the total combinations: \[ P(\text{both red}) = \frac{\text{Red combinations}}{\text{Total combinations}} = \frac{3}{36} = \frac{1}{12} \] ### **Final Answer:** The probability that both balls are red is \(\frac{1}{12}\).