Question
A bag contains eight red marbles and six blue marbles. You randomly pick a marble and then return it to the bag before picking another marble. The first marble is red and the second marble is blue. Determine whether the scenario involves independent or dependent events. Then find the probability.
(1 Point)
Dependent; 0.308
Independent; 0.04
Independent; 0.245
Independent; 0.111
(1 Point)
Dependent; 0.308
Independent; 0.04
Independent; 0.245
Independent; 0.111
Answers
GPT-4o mini
To determine whether the events are independent or dependent, we first need to clarify the definitions:
- **Independent Events**: Two events are independent if the occurrence of one does not affect the occurrence of the other.
- **Dependent Events**: Two events are dependent if the occurrence of one event affects the probability of the other event.
In this scenario, we are picking a marble, replacing it, and then picking another marble. Since the first marble is returned to the bag before the second marble is picked, the events are independent. The outcome of the first pick does not affect the outcome of the second pick due to the replacement.
Now, let's calculate the probability.
1. The probability of picking a red marble first:
\[
P(\text{Red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{8}{14} = \frac{4}{7}
\]
2. The probability of picking a blue marble second (since the first one was replaced, the composition of the bag remains the same):
\[
P(\text{Blue}) = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{6}{14} = \frac{3}{7}
\]
3. Since the events are independent, the joint probability is:
\[
P(\text{Red first and Blue second}) = P(\text{Red}) \times P(\text{Blue}) = \left(\frac{4}{7}\right) \times \left(\frac{3}{7}\right) = \frac{12}{49}
\]
Calculating \( \frac{12}{49} \) approximately results in:
\[
\frac{12}{49} \approx 0.2449
\]
Based on the options provided, the closest decimal value is \( 0.245 \).
**Thus, the events are independent, and the probability is approximately \( 0.245 \).**
Final answer: **Independent; 0.245**.
- **Independent Events**: Two events are independent if the occurrence of one does not affect the occurrence of the other.
- **Dependent Events**: Two events are dependent if the occurrence of one event affects the probability of the other event.
In this scenario, we are picking a marble, replacing it, and then picking another marble. Since the first marble is returned to the bag before the second marble is picked, the events are independent. The outcome of the first pick does not affect the outcome of the second pick due to the replacement.
Now, let's calculate the probability.
1. The probability of picking a red marble first:
\[
P(\text{Red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{8}{14} = \frac{4}{7}
\]
2. The probability of picking a blue marble second (since the first one was replaced, the composition of the bag remains the same):
\[
P(\text{Blue}) = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{6}{14} = \frac{3}{7}
\]
3. Since the events are independent, the joint probability is:
\[
P(\text{Red first and Blue second}) = P(\text{Red}) \times P(\text{Blue}) = \left(\frac{4}{7}\right) \times \left(\frac{3}{7}\right) = \frac{12}{49}
\]
Calculating \( \frac{12}{49} \) approximately results in:
\[
\frac{12}{49} \approx 0.2449
\]
Based on the options provided, the closest decimal value is \( 0.245 \).
**Thus, the events are independent, and the probability is approximately \( 0.245 \).**
Final answer: **Independent; 0.245**.