A bag contains 5 balls of unknown colours. A ball is drawn at random from it and is found to be red. Then the probability that all tha balls in the bag are red, is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(2)/(5)`
D. `(1)/(3)`

Answer

To solve the problem, we need to find the probability that all the balls in the bag are red, given that we have drawn one red ball. We can use Bayes' theorem for this purpose.

### Step-by-Step Solution:

1. **Define Events:**
- Let \( A \) be the event that all 5 balls are red.
- Let \( B \) be the event that we draw a red ball.

2. **Calculate the Probability of \( A \):**
- The probability that all balls are red, \( P(A) \), is \( \frac{1}{5} \) because there are 5 possible configurations of colors for the balls (1 red, 2 red, 3 red, 4 red, and 5 red).

3. **Calculate the Probability of \( B \) given \( A \):**
- If all balls are red, the probability of drawing a red ball, \( P(B|A) \), is \( 1 \) (since all balls are red).

4. **Calculate the Probability of \( B \) given other configurations:**
- If there are 4 red balls and 1 non-red ball, \( P(B|4 \text{ red}) = \frac{4}{5} \).
- If there are 3 red balls and 2 non-red balls, \( P(B|3 \text{ red}) = \frac{3}{5} \).
- If there are 2 red balls and 3 non-red balls, \( P(B|2 \text{ red}) = \frac{2}{5} \).
- If there is 1 red ball and 4 non-red balls, \( P(B|1 \text{ red}) = \frac{1}{5} \).

5. **Calculate Total Probability of \( B \):**
- We can calculate \( P(B) \) using the law of total probability:
\[ P(B) = P(B|A)P(A) + P(B|4 \text{ red})P(4 \text{ red}) + P(B|3 \text{ red})P(3 \text{ red}) + P(B|2 \text{ red})P(2 \text{ red}) + P(B|1 \text{ red})P(1 \text{ red}) \]
- Assuming each configuration is equally likely, we have:
\[ P(B) = 1 \cdot \frac{1}{5} + \frac{4}{5} \cdot \frac{1}{5} + \frac{3}{5} \cdot \frac{1}{5} + \frac{2}{5} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{1}{5} \]
- This simplifies to:
\[ P(B) = \frac{1}{5} + \frac{4}{25} + \frac{3}{25} + \frac{2}{25} + \frac{1}{25} = \frac{1}{5} + \frac{10}{25} = \frac{1}{5} + \frac{2}{5} = \frac{3}{5} \]

6. **Apply Bayes' Theorem:**
- Now we can find \( P(A|B) \):
\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{1 \cdot \frac{1}{5}}{\frac{3}{5}} = \frac{1}{3} \]

### Final Answer:
Thus, the probability that all the balls in the bag are red, given that we have drawn one red ball, is \( \frac{1}{3} \).