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.