A bag contains 5 black balls and 4 white balls. If I draw out three balls at random, what is the chance that they will be of same colour?

Options

A. `(1)/(8)`
B. `(1)/(6)`
C. `(3)/(8)`
D. `(5)/(6)`

Answer

To solve the problem of finding the probability that all three balls drawn from a bag containing 5 black balls and 4 white balls are of the same color, we will follow these steps:

### Step 1: Determine the total number of balls
The bag contains:
- 5 black balls
- 4 white balls

Total number of balls = 5 + 4 = 9

### Step 2: Calculate the total number of ways to choose 3 balls from 9
We can use the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) to find the total number of ways to choose 3 balls from 9.

\[ \text{Total ways to choose 3 balls} = C(9, 3) = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 3: Calculate the number of ways to choose 3 balls of the same color
**Case 1: All 3 balls are black**
\[ \text{Ways to choose 3 black balls} = C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

**Case 2: All 3 balls are white**
\[ \text{Ways to choose 3 white balls} = C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4}{1} = 4 \]

### Step 4: Calculate the total ways to choose 3 balls of the same color
Total ways to choose 3 balls of the same color = Ways to choose 3 black balls + Ways to choose 3 white balls

\[ \text{Total same color ways} = 10 + 4 = 14 \]

### Step 5: Calculate the probability
The probability that all three balls drawn are of the same color is given by the ratio of the number of favorable outcomes to the total outcomes.

\[ P(\text{same color}) = \frac{\text{Total same color ways}}{\text{Total ways to choose 3 balls}} = \frac{14}{84} = \frac{1}{6} \]

### Final Answer
The probability that all three balls drawn are of the same color is \( \frac{1}{6} \).

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