Discuss the differences in your answers to​ (a) and​ (b). Choose the correct answer below.

A.

The outcomes in​ (a) are collectively​ exhaustive, while the outcomes in​ (b) are mutually exclusive.

B.

There are two mutually exclusive and collectively exhaustive outcomes in​ (a) and

sixsix

in​ (b).

C.

The outcomes in​ (a) are not mutually exclusive or collectively​ exhaustive, while the outcomes in​ (b) are mutually exclusive and collectively exhaustive.

D.

The outcomes in​ (a) are mutually​ exclusive, while the outcomes in​ (b) are collectively exhaustive.

To properly differentiate between options (a) and (b), we must understand the concepts of mutually exclusive and collectively exhaustive outcomes:

1. **Mutually Exclusive Outcomes**: These are outcomes where no two events can happen simultaneously. If one outcome occurs, the others cannot.
  
2. **Collectively Exhaustive Outcomes**: These are outcomes that cover all possible events in the scenario. In other words, no other outcomes are possible besides the ones listed.

Now, let’s evaluate the choices:

**Choice A:**  
"The outcomes in (a) are collectively exhaustive, while the outcomes in (b) are mutually exclusive."  
- This suggests that in scenario (a), all possible outcomes are accounted for (collectively exhaustive), but in scenario (b), no two outcomes can happen at the same time (mutually exclusive).  
- This option is possible, but we need more information about the exact nature of the outcomes in both scenarios to confirm it.

**Choice B:**  
"There are two mutually exclusive and collectively exhaustive outcomes in (a) and six in (b)."  
- This states that in (a), there are only two outcomes, which are mutually exclusive and collectively exhaustive. In (b), there are six possible outcomes.
- If this aligns with the scenarios, it could be correct, but more specific details are required.

**Choice C:**  
"The outcomes in (a) are not mutually exclusive or collectively exhaustive, while the outcomes in (b) are mutually exclusive and collectively exhaustive."
- This would mean that in (a), the outcomes overlap (not mutually exclusive) and don't cover all possibilities (not collectively exhaustive), while in (b), the outcomes are separate and cover all possible events.
- If the outcomes in (a) do overlap and miss covering all events, this could be the correct option.

**Choice D:**  
"The outcomes in (a) are mutually exclusive, while the outcomes in (b) are collectively exhaustive."  
- This means that in (a), no two outcomes can happen at the same time (mutually exclusive), and in (b), all possible outcomes are listed (collectively exhaustive).
- This is another plausible option, but again, specific details about the outcomes are needed to verify.

### To make the correct choice:
You need to look closely at the exact outcomes presented in (a) and (b) and assess whether they meet the definitions of mutual exclusivity and collective exhaustiveness. Based on that, you can determine which of these options best matches the situation.

If you want to dive deeper into the examples of (a) and (b), I can assist with further details or explanations.

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Would you like further clarification? Here are 5 follow-up questions to consider:

1. What are the specific outcomes in scenarios (a) and (b)?
2. How do mutually exclusive events differ from collectively exhaustive events?
3. Could a set of outcomes be neither mutually exclusive nor collectively exhaustive?
4. Can an event be mutually exclusive but not collectively exhaustive, or vice versa?
5. How does the total number of outcomes affect whether they are collectively exhaustive?

**Tip:** Remember, mutually exclusive events cannot happen at the same time, while collectively exhaustive events include every possible outcome.

Discuss the differences in your answers to​ (a) and​ (b). Choose the correct answer below.
A.
The outcomes in​ (a) are collectively​ exhaustive, while the outcomes in​ (b) are mutually exclusive.
B.
There are two mutually exclusive and collectively exhaustive outcomes in​ (a) and six in​ (b).
C.
The outcomes in​ (a) are not mutually exclusive or collectively​ exhaustive, while the outcomes in​ (b) are mutually exclusive and collectively exhaustive.
D.
The outcomes in​ (a) are mutually​ exclusive, while the outcomes in​ (b) are collectively exhaustive.

Explore the differences between mutually exclusive and collectively exhaustive outcomes with examples from scenarios (a) and (b). This probability problem is suitable for high school students in Grades 10-12 and covers core concepts of mutually exclusive and collectively exhaustive events.

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