A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons
are selected at random without replacement. Find the probability that exactly 3 of them are green.

To find the probability that exactly 3 of the 7 selected balloons are green, we can use the **hypergeometric distribution** because the selection is made **without replacement**. 

### Step 1: Define the parameters
- Total number of balloons: \( 10 + 9 + 12 + 9 = 40 \)
- Number of green balloons: \( 12 \)
- Number of non-green balloons (pink, yellow, white): \( 10 + 9 + 9 = 28 \)
- Total balloons selected: \( 7 \)
- We want exactly \( 3 \) green balloons in the selection.

### Step 2: Set up the probability formula
The hypergeometric probability formula is:

\[
P(\text{exactly } k \text{ green balloons}) = \frac{\binom{G}{k} \binom{N-G}{n-k}}{\binom{N}{n}}
\]

Where:
- \( G \) is the total number of green balloons (12),
- \( N \) is the total number of balloons (40),
- \( n \) is the number of balloons selected (7),
- \( k \) is the number of green balloons we want (3),
- \( \binom{a}{b} \) is the combination formula, \( \binom{a}{b} = \frac{a!}{b!(a-b)!} \).

### Step 3: Compute the combinations
We need three combinations:

1. \( \binom{12}{3} \): The number of ways to select 3 green balloons from 12.
2. \( \binom{28}{4} \): The number of ways to select 4 non-green balloons from 28 (since 7 - 3 = 4).
3. \( \binom{40}{7} \): The number of ways to select 7 balloons from 40.

Let's calculate them step by step.

\[
\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]
\[
\binom{28}{4} = \frac{28 \times 27 \times 26 \times 25}{4 \times 3 \times 2 \times 1} = 20475
\]
\[
\binom{40}{7} = \frac{40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 18643560
\]

### Step 4: Calculate the probability
Now we can plug these values into the formula:

\[
P(\text{exactly 3 green}) = \frac{220 \times 20475}{18643560}
\]

Let's compute this:

\[
P(\text{exactly 3 green}) = \frac{4504500}{18643560} \approx 0.2416
\]

### Final Answer:
The probability of selecting exactly 3 green balloons out of 7 is approximately \( 0.2416 \) or 24.16%.

Would you like further details on any part of the solution?

Here are 5 related questions to explore:
1. What is the probability of selecting exactly 4 green balloons?
2. How would the probability change if there were 15 green balloons?
3. What is the expected number of green balloons when selecting 7 at random?
4. How does this problem change if the balloons were selected **with replacement**?
5. How do you calculate probabilities using the **binomial distribution** instead of hypergeometric?

**Tip:** The hypergeometric distribution is best used when you're sampling without replacement, and it adjusts for a diminishing population size after each selection!

A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons are selected at random without replacement. Find the probability that exactly 3 of them are green.

Discover how to calculate the probability of selecting exactly 3 green balloons from a set of 7 drawn without replacement, using hypergeometric distribution. This problem explores key probability concepts and combinatorics, ideal for students in Grades 11-12 or early university levels.

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