Two balls are drawn, from a bag containing 3 red, 4 white and 5 black identical balls. Find the probability that they are all of the same colour.

Answer Details

There are different ways to approach this problem, but one possible method is to use combinations. First, we need to find the total number of ways to draw two balls from the bag, without replacement. This can be calculated using the formula for combinations: \(\text{Total number of ways} = \binom{12}{2} = \frac{12!}{2!10!} = 66\) Next, we need to count the number of ways to draw two balls of the same color. There are three possible cases: both red, both white, or both black. For two red balls, we can choose 2 balls from the 3 red balls in the bag, giving us: \(\text{Number of ways for two red balls} = \binom{3}{2} = 3\) Similarly, we can count the number of ways for two white balls and two black balls: \(\text{Number of ways for two white balls} = \binom{4}{2} = 6\) \(\text{Number of ways for two black balls} = \binom{5}{2} = 10\) Therefore, the total number of ways to draw two balls of the same color is: \(\text{Number of ways for two same color balls} = 3 + 6 + 10 = 19\) Finally, we can calculate the probability of drawing two balls of the same color by dividing the number of ways for two same color balls by the total number of ways: \(\text{Probability of two same color balls} = \frac{19}{66} \approx 0.288\) Therefore, the answer is, which is \(\frac{19}{66}\).