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) = 1. State the result of event A, the result of event B, and explain if the two events are independent or dependent. (4 points)

Answer 1

To create scenarios for events A and B using 6 different-color marbles, we can define event A and event B in such a way that they cover all possible outcomes when selecting the marbles.

### Scenario for Event A:
**Event A:** Selecting at least one marble of a specific color, say red, from the 6 different-color marbles.

**Result of Event A:** The outcome of event A occurs if at least one red marble is selected from the group of 6 marbles, which includes colors red, blue, green, yellow, orange, and purple.

### Scenario for Event B:
**Event B:** Selecting all 6 marbles, without any restrictions on color.

**Result of Event B:** The outcome of event B occurs if all 6 marbles are selected from the group of marbles includes red, blue, green, yellow, orange, and purple.

### Probability and Relationship:
Since selecting all 6 marbles guarantees that at least one red marble (or any other color) will be included in the selection, we have:
- \( P(A \text{ and } B) = 1 \)
This means that the occurrence of event B (selecting all 6 marbles) guarantees the occurrence of event A (redundantly ensuring at least one red marble is selected).

### Independence or Dependence:
- Events A and B are **dependent** events because the occurrence of event B directly affects the outcome of event A. Selecting all marbles ensures that at least one red marble will be selected, so the probability of event A is influenced by the occurrence of event B.

### Summary:
- **Event A:** At least one red marble is selected.
- **Event B:** All marbles are selected.
- **Relationship:** \( P(A \text{ and } B) = 1 \) indicates events A and B are dependent events.