I asked ChatGPT the following probability question:

A box contains 25 balls labelled 1 to 25. Six balls are selected at random from the box and without replacement. It is assumed all outcomes are equally likely. Determine, with clear reasoning, the probability that two balls of each colour are chosen.

It gave the following answer

To determine the probability that two balls of each color are chosen, we first need to define the colors. Let's assume that the balls labeled 1 to 8 are one color, balls labeled 9 to 16 are another color, and balls labeled 17 to 25 are a third color.

Now, we can calculate the probability step by step:

1. Calculate the total number of ways to choose 6 balls out of 25 without replacement. This can be calculated using the binomial coefficient "25 choose 6," denoted as C(25, 6).

   $C(25,6)=\frac{25!}{6!(25-6)!}=\frac{25!}{6!19!}=\frac{25\cdot 24\cdot 23\cdot 22\cdot 21\cdot 20}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}=177100$.

2. Calculate the number of ways to choose 2 balls from the first color (balls 1-8), 2 balls from the second color (balls 9-16), and 2 balls from the third color (balls 17-25). We can calculate this separately for each group and then multiply them together.

   For the first color: $C(8,2)=\frac{8!}{2!(8-2)!}=\frac{8!}{2!6!}=28$.

   For the second color: $C(8,2)=28$.

   For the third color: $C(9,2)=36$.

3. Multiply the possibilities for each color together to find the total number of ways to choose 2 balls of each color: $28\cdot 28\cdot 36=28224$.

4. Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

   Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}=\frac{28224}{177100}\approx 0.1595$.

So, the probability that two balls of each color are chosen is approximately 0.1595, or about 15.95%.

Can you see what is wrong?