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.
Answers
Answered by
GPT-4o mini
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)
1. **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.
2. **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!
### 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)
1. **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.
2. **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!