Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, five 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 has all the red ones.

To solve this problem, we need to calculate the probability that Suzan grabs all four red marbles when she selects eight marbles from the bag.

### Step 1: Total number of marbles in the bag
There are four red marbles, four green marbles, five white marbles, and three purple marbles. The total number of marbles is:

\[
4 + 4 + 5 + 3 = 16
\]

### Step 2: Total number of ways to choose 8 marbles from 16
The total number of ways to choose 8 marbles from the 16 marbles is given by the combination formula:

\[
\binom{16}{8} = \frac{16!}{8!(16-8)!} = \frac{16!}{8!8!}
\]

### Step 3: Number of ways to choose 8 marbles, including all 4 red marbles
If Suzan grabs all 4 red marbles, she needs to choose the remaining 4 marbles from the 12 non-red marbles (4 green, 5 white, 3 purple). The number of ways to choose 4 marbles from the 12 non-red ones is:

\[
\binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!}
\]

### Step 4: Probability calculation
The probability of Suzan grabbing all 4 red marbles is the ratio of the number of favorable outcomes to the total number of outcomes:

\[
\text{Probability} = \frac{\binom{12}{4}}{\binom{16}{8}}
\]

Let's compute this.

I'll calculate both combinations.The number of favorable ways to choose 4 non-red marbles (after selecting all 4 red ones) is \( 495 \), and the total number of ways to choose 8 marbles from 16 is \( 12,870 \). 

Thus, the probability of Suzan grabbing all 4 red marbles is:

\[
\frac{495}{12,870} = \frac{1}{26}
\]

So, the probability is \( \frac{1}{26} \).

Do you want more details or have any questions?

Here are 5 related questions:
1. What is the probability Suzan grabs no red marbles?
2. What is the probability Suzan grabs exactly 3 red marbles?
3. What is the probability Suzan grabs 2 or fewer red marbles?
4. How many ways can Suzan grab exactly 2 green marbles?
5. What is the total number of ways to grab at least 1 purple marble?

**Tip:** For problems involving probabilities, combinations are essential for calculating how different outcomes can occur from a set.

Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, five 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 has all the red ones.

Find out how to calculate the probability that Suzan grabs all 4 red marbles when choosing 8 marbles from a bag containing 16 marbles of different colors. Learn key combinatorics and probability concepts with step-by-step guidance. Suitable for students in Grades 9-12.

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