A purse contains 4 silver and 5 copper coins. A second purse contains 3 silver and 7 copper coins. If a coin is taken out at random from one of the purses, what is the probability that it is a copper coin ?

Answer

To find the probability of drawing a copper coin from one of the two purses, we can follow these steps:

### Step 1: Identify the total number of coins in each purse.
- **First Purse**: 4 silver coins + 5 copper coins = 9 coins
- **Second Purse**: 3 silver coins + 7 copper coins = 10 coins

### Step 2: Calculate the probability of selecting each purse.
Since there are two purses and we are selecting one at random, the probability of selecting either purse is:
- Probability of selecting the first purse (E1) = 1/2
- Probability of selecting the second purse (E2) = 1/2

### Step 3: Calculate the probability of drawing a copper coin from each purse.
- **For the first purse**:
- Number of copper coins = 5
- Total coins = 9
- Probability of drawing a copper coin given the first purse is chosen (P(Copper | E1)) = 5/9

- **For the second purse**:
- Number of copper coins = 7
- Total coins = 10
- Probability of drawing a copper coin given the second purse is chosen (P(Copper | E2)) = 7/10

### 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:
\[ P(Copper) = P(E1) \cdot P(Copper | E1) + P(E2) \cdot P(Copper | E2) \]
Substituting the values we have:
\[ P(Copper) = \left(\frac{1}{2} \cdot \frac{5}{9}\right) + \left(\frac{1}{2} \cdot \frac{7}{10}\right) \]

### Step 5: Simplify the expression.
Calculating each term:
- First term: \( \frac{1}{2} \cdot \frac{5}{9} = \frac{5}{18} \)
- Second term: \( \frac{1}{2} \cdot \frac{7}{10} = \frac{7}{20} \)

Now we need a common denominator to add these fractions. The least common multiple of 18 and 20 is 180.

Converting both fractions:
- \( \frac{5}{18} = \frac{5 \times 10}{18 \times 10} = \frac{50}{180} \)
- \( \frac{7}{20} = \frac{7 \times 9}{20 \times 9} = \frac{63}{180} \)

Now we can add them:
\[ P(Copper) = \frac{50}{180} + \frac{63}{180} = \frac{113}{180} \]

### Final Answer:
The probability that a coin drawn is a copper coin is \( \frac{113}{180} \).

---