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Let's go through the questions based on the table provided in your assignment image.

### Part (a):
We need to find the probability that a randomly chosen person is either male (M) or under 18 (\( U18 \)).

From the table:
- \( P(M) = 0.82 \)
- \( P(U18) = 0.16 \)
- \( P(M \cap U18) = 0.10 \) (i.e., the probability that a person is both male and under 18)

We use the formula for the union of two events:
\[
P(M \cup U18) = P(M) + P(U18) - P(M \cap U18)
\]
Substituting the values:
\[
P(M \cup U18) = 0.82 + 0.16 - 0.10 = 0.88
\]
Thus, the probability that a randomly chosen person is either male or under 18 is **0.88**.

### Part (c):
Here we need to find the probability that the person is neither male nor under 18. 

This is the complement of the event found in part (a), i.e., \( P((M \cup U18)^c) \), which is:
\[
P(\text{neither male nor under 18}) = 1 - P(M \cup U18) = 1 - 0.88 = 0.12
\]
Thus, the probability that a person is neither male nor under 18 is **0.12**.

### Part (d):
We need to find the probability that the person arrested is male and at least 18 years old. This is the event \( P(M \cap U18^c) \), which is given directly in the table as **0.72**.

### Part (e):
Are the events \( M \) (male) and \( U18 \) (under 18) mutually exclusive?

Two events are mutually exclusive if they cannot happen together, meaning \( P(M \cap U18) = 0 \). However, from the table, \( P(M \cap U18) = 0.10 \), which is not 0. Therefore, the events are **not mutually exclusive**.

The correct option is **E**: \( M \) and \( U18 \) are not mutually exclusive because \( P(M \cap U18) \neq 0 \).

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Would you like more details or have any other questions? Here are five related questions to expand on this topic:

1. What is the probability of a person being female and under 18 years of age?
2. How would the probabilities change if the data from the next year were included?
3. How do you calculate the probability of two independent events occurring?
4. Can two events be both mutually exclusive and independent?
5. What is the difference between the union and intersection of two events?

**Tip**: Mutually exclusive events have no overlap, meaning they cannot occur together, while independent events can occur together but do not affect each other's likelihood.

A person who was arrested in the past year is randomly chosen. What is the probability that this person is male or under the age of 18? Use the given probability table to find the answers for the different parts of the question.

Calculate the probability of a person being either male or under the age of 18 using the given probability table. Understand key concepts such as the union of events and complementary events, along with their applications in real-world scenarios. Suitable for students in Grades 9-12.

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