Question

There are eight nickels and seven dimes in your pocket. You randomly pick a coin out of your pocket and then return it to your pocket. Then you randomly pick another coin. The first coin is a nickel, and the second coin is a dime. Determine whether the scenario involves independent or dependent events. Then find the probability.
(1 Point)

Dependent; 0.267

Independent; 0.083

Independent; 0.218

Independent; 0.249

Answer 1

To determine if the events are independent or dependent, let's analyze the situation.

1. **Understanding Events**:
- You first pick a nickel, and then you pick a dime afterward, returning the coin to your pocket each time.

2. **Independence vs. Dependence**:
- Events are **independent** if the outcome of one event does not affect the outcome of the other. Conversely, they are **dependent** if the outcome of one event affects the outcome of another.
- In this scenario, since you replace the coin back in your pocket after the first pick, the outcome of the first pick (the first coin being a nickel) does not affect the outcome of the second pick (the second coin being a dime). Therefore, these events are independent.

3. **Calculating Probabilities**:
- Total coins = 8 nickels + 7 dimes = 15 coins.
- Probability of picking a nickel first:
\[
P(\text{Nickel}) = \frac{8}{15}
\]
- Probability of picking a dime second:
\[
P(\text{Dime}) = \frac{7}{15}
\]
- Since the two events are independent, the combined probability is the product of the two probabilities:
\[
P(\text{Nickel first and Dime second}) = P(\text{Nickel}) \times P(\text{Dime}) = \frac{8}{15} \times \frac{7}{15} = \frac{56}{225}
\]

4. **Calculating the Decimal Value**:
\[
\frac{56}{225} \approx 0.2489
\]

This rounds approximately to 0.249.

So, the conclusion is:

- The events are **Independent**.
- The probability of the scenario occurring is approximately **0.249**.

Thus, the correct answer is:

**Independent; 0.249**.