https://file.aiquickdraw.com/mathbot/file/1728263883284wdkcz3lw.jpg

Let's break down the problem step by step based on the image.

### Sample Space (S):
The sample space \( S \) of possible outcomes for children combinations is given as:
\[
S = \{ GB, BG, BBG, GGB, BBB, GGG \}
\]
This includes all combinations of boys (B) and girls (G) based on the condition that the couple has children until they have one boy and one girl, but no more than three children.

### Events:
- **Event A**: The couple has one boy.
    - In the sample space, the following combinations satisfy event A: \( GB, BG, GGB, BBB \) (all outcomes where at least one boy exists).
    
- **Event C**: The couple has three children.
    - This includes \( BBG, GGB, BBB, GGG \).

- **Event D**: All of the children are of the same gender.
    - This includes \( BBB \) and \( GGG \) (all outcomes where all children are either boys or girls).

### Determining if Events are Disjoint:
Two events are disjoint if they cannot happen at the same time, i.e., their intersection is empty.

1. **A and C**: 
   - Event A (having one boy) and Event C (having three children).
   - The intersection of these sets is \( BBG, GGB, BBB \), where the couple has three children and at least one boy.
   - Conclusion: **Not disjoint** (they can happen together).

2. **A and D**:
   - Event A (having one boy) and Event D (all children of the same gender).
   - The intersection would be \( BBB \) (the only case where the couple has one boy and all children are of the same gender).
   - Conclusion: **Not disjoint** (they can happen together, specifically in the case of \( BBB \)).

3. **C and D**:
   - Event C (having three children) and Event D (all children of the same gender).
   - The intersection is \( BBB \) and \( GGG \), where the couple has three children and all of them are either boys or girls.
   - Conclusion: **Not disjoint** (they can happen together).

### Summary:
- A and C: Not disjoint
- A and D: Not disjoint
- C and D: Not disjoint

Would you like more details or have any questions?

Here are five questions for further understanding:
1. How do you determine the intersection of two events in probability?
2. What does it mean for two events to be mutually exclusive (disjoint)?
3. Can two non-disjoint events be dependent on each other?
4. How do you represent events graphically using Venn diagrams?
5. How would the answer change if there were more children in the sample space?

**Tip**: Always visualize the sample space and events using Venn diagrams to easily determine if events are disjoint or overlapping.

A couple decides to have children until they have one boy and one girl, but they will not have more than three children. Determine if the given pairs of events are disjoint based on the following conditions: Event A - the couple has one boy; Event C - the couple has three children; Event D - all of the children are the same gender.

Learn how to determine if pairs of events are disjoint in the context of probability with families having up to three children. Analyze specific events, including having one boy, three children, and all children of the same gender. Suitable for high school students in Grades 10-12.

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