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Sagot :
Sure, let’s break down and answer each part of the question step-by-step.
### Part a: List the nine outcomes in the sample space.
When a commuter observes the colors of two traffic lights where each light can be red, yellow, or green, the sample space includes all possible pairs of these colors. Thus, the nine outcomes in the sample space are:
1. (red, red)
2. (red, yellow)
3. (red, green)
4. (yellow, red)
5. (yellow, yellow)
6. (yellow, green)
7. (green, red)
8. (green, yellow)
9. (green, green)
So, the sample space is:
[tex]\[ \{ (red, red), (red, yellow), (red, green), (yellow, red), (yellow, yellow), (yellow, green), (green, red), (green, yellow), (green, green) \} \][/tex]
### Part b: Let A be the event that both colors are the same. List the outcomes in A.
Event A consists of the outcomes where both traffic lights are the same color. The possible outcomes for event A are:
1. (red, red)
2. (yellow, yellow)
3. (green, green)
So, event A is:
[tex]\[ \{ (red, red), (yellow, yellow), (green, green) \} \][/tex]
### Part c: Let B be the event that the two colors are different. List the outcomes in B.
Event B consists of the outcomes where the two traffic lights are different colors. The possible outcomes for event B are:
1. (red, yellow)
2. (red, green)
3. (yellow, red)
4. (yellow, green)
5. (green, red)
6. (green, yellow)
So, event B is:
[tex]\[ \{ (red, yellow), (red, green), (yellow, red), (yellow, green), (green, red), (green, yellow) \} \][/tex]
### Part d: Let C be the event that at least one of the lights is green. List the outcomes in C.
Event C consists of the outcomes where at least one of the two traffic lights is green. The possible outcomes for event C are:
1. (red, green)
2. (yellow, green)
3. (green, red)
4. (green, yellow)
5. (green, green)
So, event C is:
[tex]\[ \{ (red, green), (yellow, green), (green, red), (green, yellow), (green, green) \} \][/tex]
### Part e: Are events A and B mutually exclusive? Explain.
Events are mutually exclusive if they cannot both occur at the same time. In other words, A and B are mutually exclusive if there are no outcomes that belong to both A and B.
- Event A: [tex]\(\{ (red, red), (yellow, yellow), (green, green) \}\)[/tex]
- Event B: [tex]\(\{ (red, yellow), (red, green), (yellow, red), (yellow, green), (green, red), (green, yellow) \}\)[/tex]
There are no common outcomes between A and B.
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have no shared outcomes, they are mutually exclusive.
### Part f: Are events A and C mutually exclusive? Explain.
Events A and C are mutually exclusive if there are no outcomes that belong to both A and C.
- Event A: [tex]\(\{ (red, red), (yellow, yellow), (green, green) \}\)[/tex]
- Event C: [tex]\(\{ (red, green), (yellow, green), (green, red), (green, yellow), (green, green) \}\)[/tex]
The outcome (green, green) is common to both sets A and C.
Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] share the outcome (green, green), they are not mutually exclusive.
### Summary
By analyzing all the conditions and outcomes carefully, we can conclude the following:
- The sample space consists of 9 outcomes.
- Event A consists of outcomes where both traffic lights are the same.
- Event B consists of outcomes where the two traffic lights are different.
- Event C consists of outcomes where at least one of the traffic lights is green.
- Events A and B are mutually exclusive.
- Events A and C are not mutually exclusive.
### Part a: List the nine outcomes in the sample space.
When a commuter observes the colors of two traffic lights where each light can be red, yellow, or green, the sample space includes all possible pairs of these colors. Thus, the nine outcomes in the sample space are:
1. (red, red)
2. (red, yellow)
3. (red, green)
4. (yellow, red)
5. (yellow, yellow)
6. (yellow, green)
7. (green, red)
8. (green, yellow)
9. (green, green)
So, the sample space is:
[tex]\[ \{ (red, red), (red, yellow), (red, green), (yellow, red), (yellow, yellow), (yellow, green), (green, red), (green, yellow), (green, green) \} \][/tex]
### Part b: Let A be the event that both colors are the same. List the outcomes in A.
Event A consists of the outcomes where both traffic lights are the same color. The possible outcomes for event A are:
1. (red, red)
2. (yellow, yellow)
3. (green, green)
So, event A is:
[tex]\[ \{ (red, red), (yellow, yellow), (green, green) \} \][/tex]
### Part c: Let B be the event that the two colors are different. List the outcomes in B.
Event B consists of the outcomes where the two traffic lights are different colors. The possible outcomes for event B are:
1. (red, yellow)
2. (red, green)
3. (yellow, red)
4. (yellow, green)
5. (green, red)
6. (green, yellow)
So, event B is:
[tex]\[ \{ (red, yellow), (red, green), (yellow, red), (yellow, green), (green, red), (green, yellow) \} \][/tex]
### Part d: Let C be the event that at least one of the lights is green. List the outcomes in C.
Event C consists of the outcomes where at least one of the two traffic lights is green. The possible outcomes for event C are:
1. (red, green)
2. (yellow, green)
3. (green, red)
4. (green, yellow)
5. (green, green)
So, event C is:
[tex]\[ \{ (red, green), (yellow, green), (green, red), (green, yellow), (green, green) \} \][/tex]
### Part e: Are events A and B mutually exclusive? Explain.
Events are mutually exclusive if they cannot both occur at the same time. In other words, A and B are mutually exclusive if there are no outcomes that belong to both A and B.
- Event A: [tex]\(\{ (red, red), (yellow, yellow), (green, green) \}\)[/tex]
- Event B: [tex]\(\{ (red, yellow), (red, green), (yellow, red), (yellow, green), (green, red), (green, yellow) \}\)[/tex]
There are no common outcomes between A and B.
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have no shared outcomes, they are mutually exclusive.
### Part f: Are events A and C mutually exclusive? Explain.
Events A and C are mutually exclusive if there are no outcomes that belong to both A and C.
- Event A: [tex]\(\{ (red, red), (yellow, yellow), (green, green) \}\)[/tex]
- Event C: [tex]\(\{ (red, green), (yellow, green), (green, red), (green, yellow), (green, green) \}\)[/tex]
The outcome (green, green) is common to both sets A and C.
Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] share the outcome (green, green), they are not mutually exclusive.
### Summary
By analyzing all the conditions and outcomes carefully, we can conclude the following:
- The sample space consists of 9 outcomes.
- Event A consists of outcomes where both traffic lights are the same.
- Event B consists of outcomes where the two traffic lights are different.
- Event C consists of outcomes where at least one of the traffic lights is green.
- Events A and B are mutually exclusive.
- Events A and C are not mutually exclusive.
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