An urn contains 8 & white balls and 10 green balls. If Vince chooses 7 balls at random from the urn, what is the probability that he will select 4 white bails and 3 green balls? Round your answer to 3 decimal places.

To solve this problem, we need to calculate the probability that Vince selects exactly 4 white balls and 3 green balls when he randomly selects 7 balls from the urn. 

Here’s the step-by-step process:

1. **Determine the total number of white and green balls:**
   - White balls: 8
   - Green balls: 10

2. **Calculate the total number of ways to choose 7 balls from the 18 balls (8 white + 10 green):**
   - This is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
   \[
   \text{Total number of ways to choose 7 balls from 18} = \binom{18}{7} = \frac{18!}{7!(18-7)!}
   \]

3. **Calculate the number of ways to choose 4 white balls out of 8:**
   \[
   \text{Number of ways to choose 4 white balls} = \binom{8}{4} = \frac{8!}{4!(8-4)!}
   \]

4. **Calculate the number of ways to choose 3 green balls out of 10:**
   \[
   \text{Number of ways to choose 3 green balls} = \binom{10}{3} = \frac{10!}{3!(10-3)!}
   \]

5. **Multiply the number of combinations for white and green balls to get the number of favorable outcomes:**
   \[
   \text{Number of favorable outcomes} = \binom{8}{4} \times \binom{10}{3}
   \]

6. **Calculate the probability:**
   \[
   P(\text{4 white, 3 green}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of ways to choose 7 balls from 18}} = \frac{\binom{8}{4} \times \binom{10}{3}}{\binom{18}{7}}
   \]

Let's compute these values.

### Step-by-step Calculation:
- \(\binom{18}{7}\)
- \(\binom{8}{4}\)
- \(\binom{10}{3}\)

Finally, the probability is:

\[
P(\text{4 white, 3 green}) = \frac{\binom{8}{4} \times \binom{10}{3}}{\binom{18}{7}}
\]

I'll compute this now.The probability that Vince will select exactly 4 white balls and 3 green balls is approximately **0.264**.

Would you like more details on how this was calculated or have any other questions?

Here are some related questions:

1. How would the probability change if Vince were to select 6 balls instead of 7?
2. What is the probability of selecting only white balls if 7 balls are chosen?
3. How many ways can Vince select exactly 5 white balls and 2 green balls?
4. If Vince selects 8 balls, what is the probability of getting 4 white balls and 4 green balls?
5. What is the total number of possible outcomes when selecting 7 balls from 18?
6. How would the probability change if there were 12 green balls instead of 10?
7. What is the probability of selecting no white balls if 7 balls are chosen?
8. How many ways can Vince select all 8 white balls?

**Tip:** Understanding how combinations work is crucial for solving problems involving selections where order does not matter.

Learn how to calculate the probability of selecting exactly 4 white balls and 3 green balls from an urn containing 8 white balls and 10 green balls. Detailed step-by-step solution using combination formulas provided. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation