A bag contains 9 marbles, 3 of which are red, 3 of which are blue, and 3 of which are yellow. Three marbles are drawn from the bag . What is the probability that they are all blue ?

Answer

To solve the problem of finding the probability that all three drawn marbles from a bag containing 9 marbles (3 red, 3 blue, and 3 yellow) are blue, we can follow these steps:

### Step 1: Identify the total number of marbles
The bag contains:
- 3 red marbles
- 3 blue marbles
- 3 yellow marbles

Total marbles = 3 (red) + 3 (blue) + 3 (yellow) = 9 marbles.

**Hint:** Always start by counting the total number of items in the set.

### Step 2: Determine the number of favorable outcomes
We want to find the probability that all three drawn marbles are blue. The number of ways to choose 3 blue marbles from the 3 available blue marbles is given by the combination formula:

\[ \text{Number of ways to choose 3 blue marbles} = \binom{3}{3} = 1 \]

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to find the number of ways to choose items.

### Step 3: Determine the total number of outcomes
Next, we need to find the total number of ways to choose any 3 marbles from the 9 marbles. This is also calculated using the combination formula:

\[ \text{Total ways to choose 3 marbles from 9} = \binom{9}{3} \]

Calculating \( \binom{9}{3} \):

\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!} \]

This simplifies to:

\[ = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \]

**Hint:** Remember to simplify your calculations step by step to avoid mistakes.

### Step 4: Calculate the probability
Now we can find the probability \( P \) that all three drawn marbles are blue:

\[ P(\text{all blue}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{84} \]

**Hint:** The probability is always the ratio of favorable outcomes to total outcomes.

### Final Answer
The probability that all three drawn marbles are blue is:

\[ \boxed{\frac{1}{84}} \]