To create a scenario involving the selection of 6 different-color marbles, let's define two events, A and B, based on specific outcomes related to the marbles.
Scenario for Event A:
Let event A represent the event of selecting 2 red marbles from a set of 6 marbles where there are, for example, 2 red marbles and 4 marbles of different colors (blue, green, yellow, and purple).
Result of Event A: The outcome of event A occurs if the selected marbles include exactly 2 red marbles.
Scenario for Event B:
Let event B represent the event of selecting at least 1 blue marble from the same set of 6 marbles.
Result of Event B: The outcome of event B occurs if at least 1 blue marble is included among the selected marbles.
Probability of Events:
For the sake of this example, we'll denote the probability of each event:
- The probability of event A, \( P(A) \): Let's assume \( P(A) = 0.5 \) for this scenario (i.e., there is a 50% chance to pick 2 red marbles).
- The probability of event B, \( P(B) \): Let's assume \( P(B) = 0.6 \) (i.e., there is a 60% chance to pick at least 1 blue marble).
Joint Probability:
To analyze the dependency of events A and B, we need \( P(A \text{ and } B) \).
Suppose through the construction of our probability calculations or through a given problem, we find that \( P(A \text{ and } B) = 0.13 \) (i.e., there is a 13% chance that we can select both 2 red marbles and at least 1 blue marble).
Independence or Dependence:
Events A and B are independent if the occurrence of one does not affect the occurrence of the other, which mathematically translates to:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
Calculating the right side:
\[ P(A) \times P(B) = 0.5 \times 0.6 = 0.30 \]
Since \( P(A \text{ and } B) = 0.13 \) does not equal \( P(A) \times P(B) = 0.30 \), this implies that events A and B are dependent. The occurrence of selecting 2 red marbles impacts the probability of selecting at least one blue marble.
Summary:
- Event A: Selecting 2 red marbles.
- Event B: Selecting at least 1 blue marble.
- \( P(A \text{ and } B) = 0.13 \)
- Events A and B are dependent.