To solve this problem, we need to calculate the probability of selecting exactly 2 red balls and 2 white balls when 4 balls are drawn randomly from an urn containing 10 red balls and 9 white balls.

Step 1: Total number of ways to select 4 balls from the urn

The total number of balls in the urn is: $10 \, \text{(red)} + 9 \, \text{(white)} = 19 \, \text{balls}$ The number of ways to choose 4 balls from 19 balls is calculated using the combination formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ So, the total number of ways to select 4 balls from 19 is: $\binom{19}{4} = \frac{19!}{4!(19-4)!} = \frac{19 \times 18 \times 17 \times 16}{4 \times 3 \times 2 \times 1} = 3876$

Step 2: Number of ways to select 2 red and 2 white balls

• The number of ways to select 2 red balls from 10 red balls is: $\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45$
• The number of ways to select 2 white balls from 9 white balls is: $\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$ Thus, the total number of ways to select 2 red balls and 2 white balls is: $45 \times 36 = 1620$

Step 3: Probability calculation

The probability of selecting exactly 2 red balls and 2 white balls is the ratio of the favorable outcomes to the total outcomes: $P(\text{2 red, 2 white}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1620}{3876}$ Calculating this: $P(\text{2 red, 2 white}) = 0.4178$

Thus, the probability of selecting exactly 2 red balls and 2 white balls is 0.4178.

Would you like to go over any step in detail?

Here are 5 questions to expand the concept:

1. What is the probability of selecting 3 red balls and 1 white ball?
2. How many ways are there to select 3 white balls and 1 red ball?
3. What is the probability of selecting 4 white balls?
4. How would the probability change if there were an equal number of red and white balls?
5. What is the probability of selecting exactly 1 red ball and 3 white balls?

Tip: When dealing with combinations, remember to break down the problem by calculating the favorable outcomes step by step before finding the probability.