If I have 10 blue balls and 20 orange balls, and I pick 7 of them, what’s the probability the sample contains three blue balls and four orange balls?

To find the probability of picking three blue balls and four orange balls out of a sample of seven, we need to use the concept of combinations.

First, let's determine the total number of possible outcomes. We have a total of 30 balls (10 blue + 20 orange), and we are picking 7 balls from that total. Therefore, the total number of possible outcomes is given by the combination formula:

\binom{n}{r} = \frac{n!}{r!(n-r)!}

where n is the total number of balls (30) and r is the number of balls we are picking (7).

The number of ways we can select 3 blue balls from 10 is given by:

\binom{10}{3} = \frac{10!}{3!(10-3)!}

Similarly, the number of ways we can select 4 orange balls from 20 is given by:

\binom{20}{4} = \frac{20!}{4!(20-4)!}

Therefore, the probability of picking three blue balls and four orange balls is the ratio of the number of favorable outcomes (selecting three blue and four orange balls) to the total number of possible outcomes.

\text{Probability}=\frac{\binom{10}{3} \cdot\binom{20}{4}}{\binom{30}{7}}=0.286

Calculating this expression will give us the final answer.