To solve this problem, we need to find the probability of getting someone in the 18–21 age bracket or someone who refused to respond. The formula to calculate the probability of two events, $A$ and $B$, where both events can happen independently or together, is:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Where:

• $P(A)$ is the probability of selecting someone in the 18–21 age bracket,
• $P(B)$ is the probability of selecting someone who refused to respond,
• $P(A \cap B)$ is the probability of selecting someone in the 18–21 age bracket and who refused to respond (since these two events can overlap).

Step 1: Calculate the total number of people.

The total number of people contacted is:

$81 \text{ (18–21 age bracket)} + 268 \text{ (22–29 age bracket)} = 349$

Step 2: Calculate $P(A)$ (the probability of selecting someone in the 18–21 age bracket).

The number of people in the 18–21 age bracket is 81, so the probability is:

$P(A) = \frac{81}{349}$

Step 3: Calculate $P(B)$ (the probability of selecting someone who refused to respond).

The number of people who refused to respond is:

$20 \text{ (18–21 age bracket)} + 26 \text{ (22–29 age bracket)} = 46$

So the probability is:

$P(B) = \frac{46}{349}$

Step 4: Calculate $P(A \cap B)$ (the probability of selecting someone in the 18–21 age bracket who also refused to respond).

The number of people in the 18–21 age bracket who refused to respond is 20, so the probability is:

$P(A \cap B) = \frac{20}{349}$

Step 5: Use the formula for $P(A \cup B)$.

$P(A \cup B) = \frac{81}{349} + \frac{46}{349} - \frac{20}{349}$

$P(A \cup B) = \frac{81 + 46 - 20}{349} = \frac{107}{349}$

Thus, the probability of getting someone in the 18–21 age bracket or someone who refused to respond is:

$P(A \cup B) = \frac{107}{349} \approx 0.3066$

So, the probability is approximately 0.3066 or 30.66%.

Would you like further details or clarifications?

Here are 5 related questions:

1. How do you calculate probabilities for independent and dependent events?
2. What is the principle behind the inclusion-exclusion rule in probability?
3. How do we determine the complement of an event in probability?
4. Can Venn diagrams help visualize probabilities for multiple events?
5. How do we extend this to more complex probability situations involving multiple overlapping events?

Tip: Always verify if events overlap when applying the formula for the union of two events to avoid double-counting.