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 they are all red ?

Answer

To solve the problem step by step, we will calculate the probability of drawing 3 red marbles from a bag containing 9 marbles (3 red, 3 blue, and 3 yellow).

### Step 1: Identify Total Outcomes
First, we need to determine the total number of ways to choose 3 marbles from the 9 marbles in the bag. This can be calculated using the combination formula:

\[ \text{Total outcomes} = \binom{9}{3} \]

### Step 2: Calculate Total Outcomes
Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\):

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

Calculating this gives:

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

### Step 3: Identify Favorable Outcomes
Next, we need to find the number of ways to choose 3 red marbles from the 3 red marbles available. This is also calculated using the combination formula:

\[ \text{Favorable outcomes} = \binom{3}{3} \]

### Step 4: Calculate Favorable Outcomes
Using the combination formula:

\[ \binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{3!}{3! \cdot 0!} = 1 \]

### Step 5: Calculate Probability
Now we can calculate the probability of drawing 3 red marbles:

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{3}{3}}{\binom{9}{3}} = \frac{1}{84} \]

### Final Answer
Thus, the probability that all three drawn marbles are red is:

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

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