One bag contains 3 white and 2 black balls, another bag contains 5 white and 3 black balls, if a bag is chosen at random and a ball is drawn from it, what is the chance that it is white?

Options

A. `3//8`
B. `49//80`
C. `8//13`
D. `1//2`

Answer

To solve the problem, we need to find the probability of drawing a white ball from either of the two bags. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the contents of each bag
- **Bag 1** contains 3 white balls and 2 black balls.
- **Bag 2** contains 5 white balls and 3 black balls.

### Step 2: Calculate the total number of balls in each bag
- **Total in Bag 1** = 3 white + 2 black = 5 balls
- **Total in Bag 2** = 5 white + 3 black = 8 balls

### Step 3: Determine the probability of selecting each bag
Since a bag is chosen at random, the probability of selecting either bag is:
- \( P(\text{Bag 1}) = \frac{1}{2} \)
- \( P(\text{Bag 2}) = \frac{1}{2} \)

### Step 4: Calculate the probability of drawing a white ball from each bag
- **For Bag 1**:
- Probability of drawing a white ball from Bag 1:
\[ P(W | \text{Bag 1}) = \frac{\text{Number of white balls in Bag 1}}{\text{Total number of balls in Bag 1}} = \frac{3}{5} \]

- **For Bag 2**:
- Probability of drawing a white ball from Bag 2:
\[ P(W | \text{Bag 2}) = \frac{\text{Number of white balls in Bag 2}}{\text{Total number of balls in Bag 2}} = \frac{5}{8} \]

### 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 can be calculated as:
\[ P(W) = P(W | \text{Bag 1}) \cdot P(\text{Bag 1}) + P(W | \text{Bag 2}) \cdot P(\text{Bag 2}) \]

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

### Step 6: Calculate each term
Calculating the first term:
\[ \frac{3}{5} \cdot \frac{1}{2} = \frac{3}{10} \]

Calculating the second term:
\[ \frac{5}{8} \cdot \frac{1}{2} = \frac{5}{16} \]

### Step 7: Find a common denominator and add the fractions
The common denominator for 10 and 16 is 80. We convert each fraction:
- Convert \( \frac{3}{10} \) to have a denominator of 80:
\[ \frac{3}{10} = \frac{3 \times 8}{10 \times 8} = \frac{24}{80} \]

- Convert \( \frac{5}{16} \) to have a denominator of 80:
\[ \frac{5}{16} = \frac{5 \times 5}{16 \times 5} = \frac{25}{80} \]

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

### Final Answer
Thus, the probability that the drawn ball is white is:
\[ \boxed{\frac{49}{80}} \]