An urn contains 10 red balls and 9 white balls. If you are to randomly select four balls from the urn, then what is the probability that you selected two red and two white balls? Round your answer to four decimal places.


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.

An urn contains 10 red balls and 9 white balls. If you are to randomly select four balls from the urn, then what is the probability that you selected two red and two white balls? Round your answer to four decimal places.

Calculate the probability of selecting 2 red and 2 white balls from an urn containing 10 red and 9 white balls. This problem covers key probability and combinatorics concepts, using the combination formula. Suitable for students in Grades 10-12.

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