Let's tackle the problem step by step.
### Step 1: Determine the total number of coins in the bag.
The bag contains:
- 5 quarters
- 2 nickels
- 3 dimes
Adding these, we get:
[tex]\[ 5 + 2 + 3 = 10 \][/tex]
So, the total number of coins in the bag is 10.
### Step 2: Calculate the probability of selecting a quarter.
The probability of selecting a quarter is the number of quarters divided by the total number of coins:
[tex]\[ P(\text{Quarter}) = \frac{\text{Number of quarters}}{\text{Total number of coins}} = \frac{5}{10} = \frac{1}{2} \][/tex]
### Step 3: Calculate the probability of selecting a nickel.
The probability of selecting a nickel is the number of nickels divided by the total number of coins:
[tex]\[ P(\text{Nickel}) = \frac{\text{Number of nickels}}{\text{Total number of coins}} = \frac{2}{10} = \frac{1}{5} \][/tex]
### Step 4: Calculate the probability of selecting a quarter followed by a nickel.
Since the coin is replaced after the first selection, the draw of the second coin is independent of the first draw. Therefore, the combined probability of both events happening (selecting a quarter first and then a nickel) is the product of their individual probabilities:
[tex]\[ P(\text{Quarter then Nickel}) = P(\text{Quarter}) \times P(\text{Nickel}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{5}\right) = \frac{1}{10} \][/tex]
### Conclusion
The probability of selecting a quarter followed by a nickel is:
[tex]\[ \frac{1}{10} \][/tex]
This is the final answer.
