To create a scenario involving the selection of 6 different-color marbles, let's define events A and B clearly.
Definitions of Events
- Total Marbles: 6 different colors of marbles.
- Let’s denote the different colors as: Red, Blue, Green, Yellow, Orange, and Purple.
Scenario for Event A
Event A: Selecting at least one marble of the color Blue.
Result of Event A: This event occurs if at least one of the selected marbles is Blue.
Scenario for Event B
Event B: Selecting all 6 colors of marbles.
Result of Event B: This event occurs if all selected marbles include Red, Blue, Green, Yellow, Orange, and Purple, with no missing colors.
Probability Considerations
To simplify the probability calculation, we need to specify the total number of possible outcomes when selecting 6 marbles. Since there are 6 different colors, if we consider the selection as distinct events, there are:
- Total Outcomes for selecting 6 marbles: The total number of ways to select all 6 different marbles is given by the number of permutations of 6 unique items, which is \(6! = 720\).
Calculation of P(A and B)
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Event A (At least one Blue): For there to be at least one Blue marble, and for Event B to also occur (which is selecting all colors), the scenario needs to be that among the 6 marbles, all must be selected — which includes the Blue marble.
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Event B (All colors selected): This can only occur if each color is selected exactly once. Hence, both events A and B will definitely occur together (since if we select all colors, we are inherently including at least one Blue).
So mathematically:
- P(A and B): Probability that we select all the colors including Blue when selecting exactly 6 marbles = The probability of selecting all 6 colors = \( \frac{1}{720} \).
However, you mentioned \(P(A \text{ and } B) = 130\), which seems like a misunderstanding because a probability value cannot exceed 1.
Independence or Dependence
- Independent Events: Two events A and B are independent if the occurrence of A does not affect the occurrence of B (i.e., \(P(A|B) = P(A)\)).
- Dependent Events: Two events are dependent if the occurrence of one affects the occurrence of the other (i.e., \(P(A|B) \neq P(A)\)).
In this case:
- Dependence: The event A (at least one Blue marble is selected) depends on event B (all colors being selected) because if all colors are selected, at least one Blue must also be included. Thus, we conclude that A and B are dependent events.
Summary
- Event A: Selecting at least one Blue marble.
- Event B: Selecting all 6 colors of marbles.
- Dependence: Events A and B are dependent events because the occurrence of one affects the occurrence of the other.
Note: The value \(P(A \text{ and } B) = 130\) needs clarification. It cannot be a probability value as probabilities range from 0 to 1. If you meant a different kind of calculation or scenario, please specify!
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