To create a scenario with two events A and B, let's first define them clearly:
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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.
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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.
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Total Possible Outcomes: The total number of ways to choose 5 marbles from the 6 available colors is given by the combination formula \( C(n, k) \): \[ C(6, 5) = \frac{6!}{5!(6-5)!} = 6 \]
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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: \[ P(A) = \frac{1}{6} \]
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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: \[ P(B) = \frac{1}{6} \]
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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: \[ P(A \text{ and } B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \]
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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.
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