A bag contains 13 white and 7 black balls. Two are drawn at random . What is the probability that they are of the same colour ?

Options

A. `(41)/(190)`
B. `(21)/(190)`
C. `(59)/(190)`
D. `(99)/(190)`

Explanation

To solve the problem of finding the probability that two balls drawn from a bag containing 13 white balls and 7 black balls are of the same color, we can follow these steps:

### Step 1: Determine the total number of balls
In the bag, there are:
- 13 white balls
- 7 black balls

Total number of balls = 13 + 7 = 20

### Step 2: Calculate the total possible outcomes for drawing 2 balls
The total number of ways to choose 2 balls from 20 can be calculated using the combination formula:
\[ \text{Total outcomes} = \binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190 \]

### Step 3: Calculate the favorable outcomes for drawing 2 balls of the same color
We need to consider two cases for the favorable outcomes:
1. Both balls are white
2. Both balls are black

**Case 1: Both balls are white**
The number of ways to choose 2 white balls from 13:
\[ \text{Favorable outcomes (white)} = \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 \]

**Case 2: Both balls are black**
The number of ways to choose 2 black balls from 7:
\[ \text{Favorable outcomes (black)} = \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 4: Calculate the total favorable outcomes
Now, we add the favorable outcomes from both cases:
\[ \text{Total favorable outcomes} = 78 + 21 = 99 \]

### Step 5: Calculate the probability
The probability that both balls drawn are of the same color is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{99}{190} \]

### Final Answer
The probability that both balls drawn are of the same color is \(\frac{99}{190}\).

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