A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that atleast one of the three marbles drawn be black, if the first marble is red?

Options

A. `25/56`
B. `21/56`
C. `19/56`
D. `27/56`

Explanation

To solve the problem, we need to find the probability that at least one of the three marbles drawn is black, given that the first marble drawn is red.

### Step-by-Step Solution:

1. **Understand the Total Marbles**:
- There are 5 red marbles and 3 black marbles in the bag.
- Total marbles = 5 + 3 = 8.

2. **Given Condition**:
- The first marble drawn is red.
- After this draw, we have 4 red marbles and 3 black marbles left in the bag.

3. **Define the Event**:
- We need to find the probability of drawing at least one black marble in the next two draws (the second and third draws).

4. **Calculate the Complement**:
- It is often easier to calculate the probability of the complement event, which is that both of the next two marbles drawn are red.
- If both the second and third marbles are red, we can calculate that probability.

5. **Calculate the Probability of Drawing Two Red Marbles**:
- The probability of drawing a red marble on the second draw:
\[ P(\text{2nd marble is red}) = \frac{4}{7} \]
- After drawing one red marble, we have 3 red and 3 black marbles left.
- The probability of drawing a red marble on the third draw:
\[ P(\text{3rd marble is red | 2nd marble is red}) = \frac{3}{6} = \frac{1}{2} \]

6. **Combine the Probabilities**:
- The probability of both the second and third marbles being red is:
\[ P(\text{2nd red and 3rd red}) = P(\text{2nd red}) \times P(\text{3rd red | 2nd red}) = \frac{4}{7} \times \frac{1}{2} = \frac{4}{14} = \frac{2}{7} \]

7. **Calculate the Probability of At Least One Black Marble**:
- The probability of at least one black marble being drawn in the last two draws is the complement of both being red:
\[ P(\text{at least one black}) = 1 - P(\text{both red}) = 1 - \frac{2}{7} = \frac{5}{7} \]

8. **Final Calculation**:
- The total probability that at least one of the three marbles drawn is black, given that the first marble is red, is:
\[ P(\text{at least one black}) = \frac{5}{7} \]

### Final Answer:
The probability that at least one of the three marbles drawn is black, given that the first marble is red, is \(\frac{5}{7}\).