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Let's break down the table and answer the probability questions:

### Table Breakdown:
- The table shows pet preferences categorized by the owner's age:
  
|                | Cats | Dogs | Other | Total |
|----------------|------|------|-------|-------|
| **Over 40**    | 100  | 50   | 30    | ?     |
| **40 or Under**| 50   | 50   | 20    | ?     |
| **Total**      | ?    | ?    | ?     | ?     |

First, let's calculate the totals for each row and column:
- **Total for Over 40** = \( 100 + 50 + 30 = 180 \)
- **Total for 40 or Under** = \( 50 + 50 + 20 = 120 \)
- **Grand total (all people)** = \( 180 + 120 = 300 \)

Now, the totals for each pet type:
- **Total Cats** = \( 100 + 50 = 150 \)
- **Total Dogs** = \( 50 + 50 = 100 \)
- **Total Other** = \( 30 + 20 = 50 \)

### Now for the probabilities:

#### Part a) Probability that a person is "Over 40" *or* owns an "Other" pet:

- People "Over 40" = 180
- People who own "Other" pets = \( 30 + 20 = 50 \)
- Overlap (people "Over 40" *and* own an "Other" pet) = 30

Using the formula for the probability of \( A \cup B \):
\[
P(A \cup B) = \frac{|A| + |B| - |A \cap B|}{\text{Total}}
\]
\[
P(\text{Over 40 or Other}) = \frac{180 + 50 - 30}{300} = \frac{200}{300} = \frac{2}{3}
\]

#### Part b) Probability that a person owns a cat *or* owns an "Other" pet:

- People who own Cats = 150
- People who own "Other" pets = 50
- Overlap (people who own both Cats *and* "Other" pets) = 0 (since no overlap is indicated)

So, using the formula again:
\[
P(\text{Cat or Other}) = \frac{150 + 50 - 0}{300} = \frac{200}{300} = \frac{2}{3}
\]

### Final Answers:
- a) Probability = \( \frac{2}{3} \)
- b) Probability = \( \frac{2}{3} \)

Let me know if you'd like more details or further explanation! 

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Here are 5 follow-up questions you could explore:
1. What is the probability of selecting a person who is under 40 and owns a dog?
2. What is the probability of selecting a person who is under 40 and owns a pet of any kind?
3. What is the probability of selecting someone who is over 40 and does not own a dog?
4. How would the probabilities change if the total number of people increased by 10%?
5. What is the probability that a randomly selected person owns only a dog?

**Tip:** Always remember to use the inclusion-exclusion principle when calculating probabilities for "or" conditions!

A person is selected at random. Determine the probability that: a) They are 'Over 40' or they own an 'Other' pet. b) They own a cat or they own an 'Other' pet.

Determine the probability that a person selected at random is either 'Over 40' or owns an 'Other' pet, using the inclusion-exclusion principle. Explore key probability concepts with this pet ownership problem, ideal for high school students.

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