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)`

Explanation

To solve the problem step by step, we will calculate the probability of drawing a white ball after randomly selecting one of the two bags.

### 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 choosing each bag
Since there are 2 bags and one is chosen at random:
- Probability of choosing Bag 1 = \( \frac{1}{2} \)
- Probability of choosing Bag 2 = \( \frac{1}{2} \)

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

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

### 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 follows:
\[ P(\text{White}) = P(\text{Bag 1}) \cdot P(\text{White | Bag 1}) + P(\text{Bag 2}) \cdot P(\text{White | Bag 2}) \]
Substituting the values we calculated:
\[ P(\text{White}) = \left(\frac{1}{2} \cdot \frac{3}{5}\right) + \left(\frac{1}{2} \cdot \frac{5}{8}\right) \]

### Step 6: Simplify the expression
Calculating each term:
- For Bag 1:
\[ \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \]
- For Bag 2:
\[ \frac{1}{2} \cdot \frac{5}{8} = \frac{5}{16} \]

Now we need a common denominator to add these two fractions. The least common multiple of 10 and 16 is 80.
- 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 add the two fractions:
\[ P(\text{White}) = \frac{24}{80} + \frac{25}{80} = \frac{49}{80} \]

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

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