Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, two white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms.
She does not have all the green ones.

We are asked to find the probability that Suzan grabs 8 marbles from a bag containing a total of 12 marbles, but does not grab all 3 green ones.

### Step 1: Total number of marbles
The bag contains:
- 4 red marbles,
- 3 green marbles,
- 2 white marbles,
- 3 purple marbles.

Thus, the total number of marbles is:
\[
4 + 3 + 2 + 3 = 12 \text{ marbles}.
\]

### Step 2: Total number of possible ways to choose 8 marbles
Suzan selects 8 marbles out of 12. The number of ways to choose 8 marbles from 12 is given by the combination formula:
\[
\binom{12}{8} = \frac{12!}{8!(12-8)!} = \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495.
\]

### Step 3: Number of favorable outcomes (does not grab all green ones)
We first calculate the number of unfavorable outcomes, where Suzan grabs all 3 green marbles. In this case, she selects 3 green marbles and 5 other marbles from the remaining 9 marbles (since 12 - 3 = 9).

The number of ways to choose 5 marbles from the remaining 9 marbles (non-green) is:
\[
\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1} = 126.
\]

Therefore, the number of ways to grab all 3 green marbles is 126.

### Step 4: Number of favorable outcomes
The total number of favorable outcomes is the total number of outcomes minus the number of unfavorable outcomes (grabbing all green marbles). So:
\[
\text{Favorable outcomes} = 495 - 126 = 369.
\]

### Step 5: Probability of not grabbing all green marbles
The probability is the ratio of favorable outcomes to total outcomes:
\[
P(\text{not all green}) = \frac{369}{495}.
\]

Now, simplify the fraction:
\[
\frac{369}{495} = \frac{123}{165} = \frac{41}{55}.
\]

Thus, the probability that Suzan does not grab all the green marbles is:
\[
\boxed{\frac{41}{55}}.
\]

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

Here are 5 related questions to expand your understanding:
1. How would you calculate the probability that Suzan grabs exactly 2 green marbles?
2. What is the probability that she grabs no red marbles?
3. If she grabs 4 purple marbles, what are the chances that she also grabs at least 1 white marble?
4. How does the probability change if the bag had more than 12 marbles?
5. What is the difference in approach if Suzan grabs marbles without replacement?

**Tip:** When calculating probabilities with combinations, always break the problem into favorable and total outcomes for easier calculations!

Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, two white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. She does not have all the green ones.

Calculate the probability that Suzan, grabbing 8 marbles from a bag with different colored marbles, does not grab all the green ones. Explore combinatorial methods and probability concepts with a detailed step-by-step solution. Suitable for students in Grades 10-12.

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