A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins and 2 silver coins. A coin is taken out of any purse, the probability that it is a copper coin is

Options

A. `4/7`
B. `3/4`
C. `3/7`
D. `(37)/(56)`

Answer

To solve the problem of finding the probability that a coin taken out of either purse is a copper coin, we can follow these steps:

### Step 1: Understand the contents of each purse
- **Purse 1** contains:
- 4 copper coins
- 3 silver coins
- Total = 4 + 3 = 7 coins

- **Purse 2** contains:
- 6 copper coins
- 2 silver coins
- Total = 6 + 2 = 8 coins

### Step 2: Determine the probability of selecting each purse
Since there are two purses, the probability of selecting either purse is:
- Probability of selecting Purse 1, P(Purse 1) = 1/2
- Probability of selecting Purse 2, P(Purse 2) = 1/2

### Step 3: Calculate the probability of drawing a copper coin from each purse
- For **Purse 1**:
- Probability of drawing a copper coin from Purse 1, P(Copper | Purse 1) = Number of copper coins in Purse 1 / Total coins in Purse 1
- P(Copper | Purse 1) = 4 / 7

- For **Purse 2**:
- Probability of drawing a copper coin from Purse 2, P(Copper | Purse 2) = Number of copper coins in Purse 2 / Total coins in Purse 2
- P(Copper | Purse 2) = 6 / 8 = 3 / 4

### Step 4: Use the law of total probability to find the overall probability of drawing a copper coin
The total probability of drawing a copper coin, P(Copper), can be calculated as follows:
\[ P(Copper) = P(Purse 1) \cdot P(Copper | Purse 1) + P(Purse 2) \cdot P(Copper | Purse 2) \]
Substituting the values:
\[ P(Copper) = \left(\frac{1}{2} \cdot \frac{4}{7}\right) + \left(\frac{1}{2} \cdot \frac{3}{4}\right) \]

### Step 5: Simplify the expression
Calculating each term:
1. For Purse 1:
\[ \frac{1}{2} \cdot \frac{4}{7} = \frac{4}{14} = \frac{2}{7} \]

2. For Purse 2:
\[ \frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8} \]

Now combine these:
\[ P(Copper) = \frac{2}{7} + \frac{3}{8} \]

### Step 6: Find a common denominator and add the fractions
The least common multiple of 7 and 8 is 56. Convert each fraction:
\[ \frac{2}{7} = \frac{2 \times 8}{7 \times 8} = \frac{16}{56} \]
\[ \frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56} \]

Now add them:
\[ P(Copper) = \frac{16}{56} + \frac{21}{56} = \frac{37}{56} \]

### Final Answer
The probability that a coin taken out of either purse is a copper coin is:
\[ \frac{37}{56} \]

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