A purse contains 5 silver and 2 gold coins. A second purse contains 4 silver and 3 gold coins. A coin is taken out of any purse. The probability that it is a silver coin is

Options

A. `9/6`
B. `20/40`
C. `9/14`
D. None of these

Answer

To solve the problem, we need to find the probability of selecting a silver coin from two purses. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the contents of each purse
- **Purse 1** contains:
- 5 silver coins
- 2 gold coins
- **Purse 2** contains:
- 4 silver coins
- 3 gold coins

### Step 2: Calculate the total number of coins in each purse
- **Total coins in Purse 1** = 5 silver + 2 gold = 7 coins
- **Total coins in Purse 2** = 4 silver + 3 gold = 7 coins

### Step 3: Define the events
- Let **E1** be the event of selecting Purse 1.
- Let **E2** be the event of selecting Purse 2.
- Let **A** be the event of selecting a silver coin.

### Step 4: Calculate the probabilities of selecting each purse
- The probability of selecting Purse 1, \( P(E1) \) = \( \frac{1}{2} \)
- The probability of selecting Purse 2, \( P(E2) \) = \( \frac{1}{2} \)

### Step 5: Calculate the probability of selecting a silver coin from each purse
- From **Purse 1**:
- Probability of selecting a silver coin, \( P(A|E1) \) = \( \frac{5 \text{ silver coins}}{7 \text{ total coins}} = \frac{5}{7} \)

- From **Purse 2**:
- Probability of selecting a silver coin, \( P(A|E2) \) = \( \frac{4 \text{ silver coins}}{7 \text{ total coins}} = \frac{4}{7} \)

### Step 6: Use the law of total probability to find \( P(A) \)
The total probability of selecting a silver coin can be calculated using the formula:
\[ P(A) = P(E1) \cdot P(A|E1) + P(E2) \cdot P(A|E2) \]

Substituting the values we calculated:
\[ P(A) = \left(\frac{1}{2} \cdot \frac{5}{7}\right) + \left(\frac{1}{2} \cdot \frac{4}{7}\right) \]

### Step 7: Simplify the expression
\[ P(A) = \frac{1}{2} \left(\frac{5}{7} + \frac{4}{7}\right) = \frac{1}{2} \cdot \frac{9}{7} = \frac{9}{14} \]

### Final Answer
The probability that a coin taken out is a silver coin is \( \frac{9}{14} \).