A bag contains 3 white balls and 2 black balls, another contains 5 white and 3 black balls, if a bag is chosen at random and a ball is drawn from it. What is the probability that it is white ?

Answer

To solve the problem step by step, we will calculate the probability of drawing a white ball from the two bags.

### Step 1: Identify the contents of the bags.
- Bag A contains 3 white balls and 2 black balls.
- Bag B contains 5 white balls and 3 black balls.

### Step 2: Calculate the total number of balls in each bag.
- Total balls in Bag A = 3 (white) + 2 (black) = 5 balls.
- Total balls in Bag B = 5 (white) + 3 (black) = 8 balls.

### Step 3: Calculate the probability of choosing each bag.
Since there are 2 bags and a bag is chosen at random:
- Probability of choosing Bag A, P(A) = 1/2.
- Probability of choosing Bag B, P(B) = 1/2.

### Step 4: Calculate the probability of drawing a white ball from each bag.
- Probability of drawing a white ball from Bag A, P(W|A) = Number of white balls in Bag A / Total number of balls in Bag A = 3/5.
- Probability of drawing a white ball from Bag B, P(W|B) = Number of white balls in Bag B / Total number of balls in Bag B = 5/8.

### Step 5: Use the law of total probability to find the overall probability of drawing a white ball.
The total probability of drawing a white ball, P(W), can be calculated as:
\[ P(W) = P(A) \cdot P(W|A) + P(B) \cdot P(W|B) \]

Substituting the values we calculated:
\[ P(W) = \left(\frac{1}{2} \cdot \frac{3}{5}\right) + \left(\frac{1}{2} \cdot \frac{5}{8}\right) \]

### Step 6: Calculate each term.
- For Bag A:
\[ P(A) \cdot P(W|A) = \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \]
- For Bag B:
\[ P(B) \cdot P(W|B) = \frac{1}{2} \cdot \frac{5}{8} = \frac{5}{16} \]

### Step 7: Add the two probabilities together.
To add \(\frac{3}{10}\) and \(\frac{5}{16}\), we need a common denominator. The least common multiple (LCM) of 10 and 16 is 80.

Convert each fraction:
- \(\frac{3}{10} = \frac{3 \times 8}{10 \times 8} = \frac{24}{80}\)
- \(\frac{5}{16} = \frac{5 \times 5}{16 \times 5} = \frac{25}{80}\)

Now add them:
\[ P(W) = \frac{24}{80} + \frac{25}{80} = \frac{49}{80} \]

### Final Answer:
The probability that the ball drawn is white is \(\frac{49}{80}\).

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