Bag A contains 3 white & 7 red balls and bag B contains 2 white & 3 Red balls. If a ball is picked up randomly then what is the probability that the ball picked is white from bag A

Options

A. `3/20`
B. `2/20`
C. `3/10`
D. `4/20`

Answer

To solve the problem, we need to find the probability that a randomly picked ball is white from Bag A. Let's break it down step by step.

### Step 1: Identify the total number of balls in each bag.
- **Bag A** contains 3 white balls and 7 red balls.
- Total balls in Bag A = 3 (white) + 7 (red) = 10 balls.
- **Bag B** contains 2 white balls and 3 red balls.
- Total balls in Bag B = 2 (white) + 3 (red) = 5 balls.

### Step 2: Calculate the probability of picking a ball from each bag.
Since we are picking a ball randomly from either bag, the probability of choosing either bag is:
- \( P(E_1) \) (probability of choosing Bag A) = \( \frac{1}{2} \)
- \( P(E_2) \) (probability of choosing Bag B) = \( \frac{1}{2} \)

### Step 3: Calculate the probability of picking a white ball from each bag.
- For **Bag A** (E1):
- Probability of picking a white ball from Bag A, \( P(A|E_1) \) = \( \frac{3}{10} \) (since there are 3 white balls out of 10 total balls).

- For **Bag B** (E2):
- Probability of picking a white ball from Bag B, \( P(A|E_2) \) = \( \frac{2}{5} \) (since there are 2 white balls out of 5 total balls).

### Step 4: Use the law of total probability to find the overall probability of picking a white ball.
The total probability of picking a white ball, \( P(A) \), is given by:
\[ P(A) = P(A|E_1) \cdot P(E_1) + P(A|E_2) \cdot P(E_2) \]
Substituting the values:
\[ P(A) = \left(\frac{3}{10} \cdot \frac{1}{2}\right) + \left(\frac{2}{5} \cdot \frac{1}{2}\right) \]
Calculating each term:
- First term: \( \frac{3}{10} \cdot \frac{1}{2} = \frac{3}{20} \)
- Second term: \( \frac{2}{5} \cdot \frac{1}{2} = \frac{2}{10} = \frac{4}{20} \)

Now, adding these two results:
\[ P(A) = \frac{3}{20} + \frac{4}{20} = \frac{7}{20} \]

### Step 5: Calculate the conditional probability that the ball picked is from Bag A given that it is white.
We want to find \( P(E_1|A) \):
\[ P(E_1|A) = \frac{P(A|E_1) \cdot P(E_1)}{P(A)} \]
Substituting the values:
\[ P(E_1|A) = \frac{\left(\frac{3}{10}\right) \cdot \left(\frac{1}{2}\right)}{\frac{7}{20}} \]
Calculating the numerator:
\[ \text{Numerator} = \frac{3}{10} \cdot \frac{1}{2} = \frac{3}{20} \]

Now substituting into the equation:
\[ P(E_1|A) = \frac{\frac{3}{20}}{\frac{7}{20}} = \frac{3}{7} \]

### Final Answer:
The probability that the ball picked is white from Bag A is \( \frac{3}{7} \).

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