2. In a box, there are 3 blue balls, 10 green balls and 12 purple balls. If 5 balls are randomly selected from the box at the same time, find (a) the probability that 5 balls of the same colour are selected; (b) the probability that at least 2 balls of different colours are selected. ( 3 marks)

To solve the problem, we first need to determine the total number of balls in the box and the total number of ways to select 5 balls from these. ### Total Number of Balls - Blue balls: 3 - Green balls: 10 - Purple balls: 12 Total number of balls = 3 + 10 + 12 = 25 balls. ### Total Ways to Select 5 Balls The total number of ways to select 5 balls from 25 is given by the combination formula \( C(n, k) \), which is defined as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] So, the total ways to select 5 balls from 25 is: \[ C(25, 5) = \frac{25!}{5!(25-5)!} = \frac{25!}{5! \cdot 20!} = \frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} = 53130 \] ### (a) Probability that 5 Balls of the Same Colour are Selected To find the probability that all 5 balls selected are of the same colour, we need to consider the possible colours: 1. **Blue Balls**: We cannot select 5 blue balls since there are only 3. 2. **Green Balls**: We can select 5 green balls. The number of ways to select 5 green balls from 10 is: \[ C(10, 5) = \frac{10!}{5!(10-5)!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \] 3. **Purple Balls**: We can select 5 purple balls. The number of ways to select 5 purple balls from 12 is: \[ C(12, 5) = \frac{12!}{5!(12-5)!} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \] Now, we add the ways to select 5 balls of the same colour: \[ \text{Total ways to select 5 balls of the same colour} = C(10, 5) + C(12, 5) = 252 + 792 = 1044 \] Now, we can find the probability: \[ P(\text{5 balls of the same colour}) = \frac{\text{Number of ways to select 5 balls of the same colour}}{\text{Total ways to select 5 balls}} = \frac{1044}{53130} \] Calculating this gives: \[ P(\text{5 balls of the same colour}) \approx 0.0197 \] ### (b) Probability that at Least 2 Balls of Different Colours are Selected To find the probability that at least 2 balls of different colours are selected, we can use the complement rule. We first find the probability that all selected balls are of the same colour (which we calculated in part (a)) and subtract it from 1. \[ P(\text{at least 2 different colours}) = 1 - P(\text{5 balls of the same colour}) = 1 - \frac{1044}{53130} \] Calculating this gives: \[ P(\text{at least 2 different colours}) = 1 - 0.0197 \approx 0.9803 \] ### Final Answers (a) The probability that 5 balls of the same colour are selected is approximately **0.0197**. (b) The probability that at least 2 balls of different colours are selected is approximately **0.9803**.