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Let's break down each question and calculate the probabilities:

### 1. **What is the probability that all the marbles are red?**

The total number of marbles in the bag is:
\[
9 \ (\text{red}) + 8 \ (\text{white}) + 6 \ (\text{blue}) = 23 \ \text{marbles}
\]

You are drawing 3 marbles without replacement, and you want all 3 to be red. The number of red marbles is 9.

The probability of drawing 3 red marbles consecutively is given by multiplying the probabilities of drawing a red marble each time, noting that after each draw, both the total number of marbles and the number of red marbles decrease.

\[
\text{P(all red)} = \frac{9}{23} \times \frac{8}{22} \times \frac{7}{21}
\]

Let's calculate that.

### 2. **What is the probability that exactly two of the marbles are red?**

We need to choose 2 red marbles and 1 non-red marble (which can be either white or blue). First, we calculate the number of ways to choose 2 red marbles and 1 non-red marble:

- The probability of drawing 2 red marbles out of 9 red ones and 1 non-red marble from the remaining 14 (8 white + 6 blue) is:
\[
\text{P(exactly 2 red)} = \left(\frac{9}{23} \times \frac{8}{22} \times \frac{14}{21}\right) + \left(\frac{9}{23} \times \frac{14}{22} \times \frac{8}{21}\right) + \left(\frac{14}{23} \times \frac{9}{22} \times \frac{8}{21}\right)
\]
This takes into account the different orders in which these marbles can be drawn.

### 3. **What is the probability that none of the marbles are red?**

This means all 3 marbles are either white or blue. There are 14 non-red marbles (8 white and 6 blue).

The probability of drawing 3 non-red marbles is:
\[
\text{P(no red)} = \frac{14}{23} \times \frac{13}{22} \times \frac{12}{21}
\]

I'll now calculate these values.Here are the calculated probabilities:

1. **The probability that all the marbles are red** is approximately **0.0474** (or 4.74%).
2. **The probability that exactly two of the marbles are red** is approximately **0.2846** (or 28.46%).
3. **The probability that none of the marbles are red** is approximately **0.2055** (or 20.55%).

Would you like further details on any of these calculations or additional explanations?

Here are 5 related questions for practice:
1. What is the probability of drawing exactly one red marble?
2. How does the probability change if there are 10 red marbles instead of 9?
3. What is the probability of drawing one red, one white, and one blue marble?
4. What is the probability of drawing three white marbles?
5. How does the probability change if we draw 4 marbles instead of 3?

**Tip:** When dealing with probabilities involving "without replacement," always consider how the total number of items and the desired subset changes with each draw.

A bag contains 9 red marbles, 8 white marbles, and 6 blue marbles. You draw 3 marbles out at random, without replacement. What is the probability that all the marbles are red? What is the probability that exactly two of the marbles are red? What is the probability that none of the marbles are red?

Learn how to calculate probabilities when drawing marbles from a bag without replacement. This problem involves finding the probability of drawing all red marbles, exactly two red marbles, or no red marbles using combinatorics and the multiplication rule for probability.

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