Question

Consider 6 different-color marbles that are randomly selected. Create a scenario for event A and a scenario for event B , such that P(A and B)=130 . State the result of event A , the result of event B , and explain if the two events are independent or dependent.

Answer 1

To create a scenario with two events A and B, let's first define them clearly:

- **Event A**: Selecting 3 specific-color marbles from a total of 6 different-color marbles. Let's say the specific colors are red, blue, and green.

- **Event B**: Selecting 2 specific-color marbles from the same total of 6, and let's say the specific colors here are yellow and purple.

### Calculation of P(A and B)

For the probability calculations, we need to calculate P(A and B) in terms of outcomes.

1. **Total Possible Outcomes**: The total number of ways to choose 5 marbles from the 6 available colors is given by the combination formula \( C(n, k) \):
\[
C(6, 5) = \frac{6!}{5!(6-5)!} = 6
\]

2. **Calculating P(A)**:
Let's assume the selection of marbles is such that Event A occurs in a way that we can define:
- For P(A) to occur, we need to select exactly the 3 marbles of colors: Red, Blue, and Green. Therefore, the favorable outcomes for Event A are just 1 way (since we need these exact colors). Thus:
\[
P(A) = \frac{1}{6}
\]

3. **Calculating P(B)**:
Similarly, for Event B (selecting exactly the 2 marbles of colors: Yellow and Purple), the favorable outcomes for Event B can also be defined in a similar way:
\[
P(B) = \frac{1}{6}
\]

4. **Calculating P(A and B)**:
For both events A and B to happen simultaneously, we need to select the specific colors (Red, Blue, Green for A and Yellow, Purple for B), which do not overlap:
Since the selection of marbles for A does not affect B:
\[
P(A \text{ and } B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}
\]

5. **However**, to match your condition of \( P(A \text{ and } B) = 130 \), we must clarify that probability cannot be greater than 1, meaning it seems there is a misunderstanding regarding the expectation of probabilistic values.

### Independence vs. Dependence

**Independence**:
In probability, events A and B are considered independent if the occurrence of one does not affect the occurrence of the other. In this case, selecting marbles for Event A (Red, Blue, Green) has no effect on selecting for Event B (Yellow, Purple). Thus:
- **Events A and B are independent**.

### Summary
- **Event A**: Selecting 3 marbles of colors Red, Blue, and Green.
- **Event B**: Selecting 2 marbles of colors Yellow and Purple.
- **P(A and B)**: Calculated as \( 130 \) does not hold; it should be \( \frac{1}{36} \).
- **Independence**: Events A and B are independent.

If you meant the scenario should yield a value of 130 in another construct (such as counts or frequencies), please clarify, and I can adjust the scenario accordingly.