An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. Probability that they are of the different colours is

Options

A. `(2)/(5)`
B. `(1)/(15)`
C. `(8)/(15)`
D. `(4)/(15)`

Explanation

To find the probability that two balls drawn from an urn containing 2 red balls and 4 black balls are of different colors, we can follow these steps:

### Step 1: Determine the total number of balls
The urn contains a total of 6 balls:
- 2 red balls
- 4 black balls

### Step 2: Calculate the total number of ways to choose 2 balls
The total number of ways to choose 2 balls from 6 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 3: Calculate the number of favorable outcomes for different colors
To find the probability that the two balls are of different colors, we can consider two scenarios:
1. The first ball is red and the second ball is black.
2. The first ball is black and the second ball is red.

**Scenario 1: First ball is red and second ball is black**
- The probability of drawing a red ball first:
\[ P(\text{Red first}) = \frac{2}{6} \]
- After drawing a red ball, there are 5 balls left (1 red and 4 black). The probability of drawing a black ball second:
\[ P(\text{Black second | Red first}) = \frac{4}{5} \]
- Therefore, the combined probability for this scenario is:
\[ P(\text{Red first and Black second}) = \frac{2}{6} \times \frac{4}{5} = \frac{8}{30} \]

**Scenario 2: First ball is black and second ball is red**
- The probability of drawing a black ball first:
\[ P(\text{Black first}) = \frac{4}{6} \]
- After drawing a black ball, there are 5 balls left (2 red and 3 black). The probability of drawing a red ball second:
\[ P(\text{Red second | Black first}) = \frac{2}{5} \]
- Therefore, the combined probability for this scenario is:
\[ P(\text{Black first and Red second}) = \frac{4}{6} \times \frac{2}{5} = \frac{8}{30} \]

### Step 4: Add the probabilities of both scenarios
Now, we can add the probabilities from both scenarios to find the total probability of drawing two balls of different colors:
\[ P(\text{Different colors}) = P(\text{Red first and Black second}) + P(\text{Black first and Red second}) = \frac{8}{30} + \frac{8}{30} = \frac{16}{30} \]

### Step 5: Simplify the probability
Now, we simplify \(\frac{16}{30}\):
\[ \frac{16}{30} = \frac{8}{15} \]

### Final Answer
Thus, the probability that the two balls drawn are of different colors is:
\[ \frac{8}{15} \]