There are 5 black and 6 red balls in a bag. If 3 balls are drawn at random, find the probability that these balls are black.

Explanation

To find the probability that all three balls drawn from a bag containing 5 black balls and 6 red balls are black, we can follow these steps:

### Step 1: Determine the total number of balls
We have:
- Black balls = 5
- Red balls = 6

Total number of balls = 5 + 6 = 11

### Step 2: Calculate the total ways to choose 3 balls from 11
The total number of ways to choose 3 balls from 11 can be calculated using the combination formula \( nCr \), which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

For our case, we need to calculate \( 11C3 \):

\[ 11C3 = \frac{11!}{3!(11-3)!} = \frac{11!}{3! \cdot 8!} \]

### Step 3: Simplify \( 11C3 \)
We can simplify \( 11C3 \):

\[ 11C3 = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Step 4: Calculate the ways to choose 3 black balls from 5
Now, we need to calculate the number of ways to choose 3 black balls from the 5 black balls, which is \( 5C3 \):

\[ 5C3 = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \]

### Step 5: Simplify \( 5C3 \)
We can simplify \( 5C3 \):

\[ 5C3 = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 6: Calculate the probability
Now, we can find the probability that all three balls drawn are black:

\[ P(\text{3 black balls}) = \frac{\text{Number of ways to choose 3 black balls}}{\text{Total ways to choose 3 balls}} = \frac{5C3}{11C3} = \frac{10}{165} \]

### Step 7: Simplify the probability
We can simplify \( \frac{10}{165} \):

\[ P(\text{3 black balls}) = \frac{2}{33} \]

### Final Answer
Thus, the probability that all three balls drawn are black is:

\[ \frac{2}{33} \]

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