https://file.aiquickdraw.com/mathbot/file/1728263655943zumvvtmk.jpg

Let's go through the questions one by one from the image.

### First Set of Events:

The family has three children. The events are:

- Event A: The middle child is a female.
- Event C: The three children are of the same sex.

#### i. Possible outcomes for Event A:
For Event A, the middle child must be female, so the possible combinations for the three children (M for male, F for female) are:

- M, **F**, M
- M, **F**, F
- F, **F**, M
- F, **F**, F

These are the four possible outcomes where the middle child is female.

#### ii. Possible outcomes for Event C:
For Event C, all three children must be of the same sex. The possible outcomes are:

- M, M, M (all male)
- F, F, F (all female)

#### iii. Overlap between events A and C:
Are there any outcomes that satisfy both Event A (middle child is female) and Event C (all children are the same sex)?

The only possible outcome where the middle child is female and all three children are of the same sex is **F, F, F**.

Thus, **yes**, there is an overlap.

#### iv. Are the events disjoint?
Since there is an overlap between Event A and Event C (the outcome F, F, F), the events are **not disjoint**.

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### Second Set of Events:

The events are:

- Event A: Exactly one of the three children is a female.
- Event C: Exactly one of the three children is a male.

#### i. Possible outcomes for Event A:
In this case, only one child is female, so the possible outcomes are:

- F, M, M
- M, F, M
- M, M, F

#### ii. Possible outcomes for Event C:
In this case, only one child is male, so the possible outcomes are:

- M, F, F
- F, M, F
- F, F, M

#### iii. Overlap between events A and C:
Are there any outcomes that satisfy both Event A (exactly one female) and Event C (exactly one male)?

No, there is no outcome that satisfies both conditions. The outcomes for Event A involve two males and one female, while the outcomes for Event C involve two females and one male.

Thus, **no**, there is no overlap.

#### iv. Are the events disjoint?
Since there is no overlap between Event A and Event C, the events are **disjoint**.

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Let me know if you'd like more details on any of these parts. Here are some related questions to further explore this topic:

1. What are the conditions for two events to be disjoint in general probability theory?
2. How would the situation change if the number of children increased to four?
3. How can a Venn diagram help visualize overlapping events?
4. What is the probability of Event A happening in each of these scenarios?
5. Can these events be mutually exclusive or independent at the same time?

**Tip:** When comparing two events, always check if they share any common outcomes to determine if they overlap.

A family has three children. Consider the following two events: A: The middle child is female; C: The three children are of the same sex. What are the possible outcomes for these events? Are they disjoint?

Explore a probability problem involving three children and gender combinations. Understand the possible outcomes for events where the middle child is female or where all children are of the same sex. Learn how to determine if events are disjoint.

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