Pollsters are concerned about declining levels of cooperation among persons contacted in surveys. A pollster contacts 81 people in the​ 18-21 age bracket and finds that 61 of them respond and 20 refuse to respond. When 268 people in the​ 22-29 age bracket are​ contacted, 242 respond and 26 refuse to respond. Suppose that one of the 349 people is randomly selected. Find the probability of getting someone in the 18 dash 21 age bracket or someone who refused to respond. 

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.

Pollsters are concerned about declining levels of cooperation among persons contacted in surveys. A pollster contacts 81 people in the 18-21 age bracket and finds that 61 of them respond and 20 refuse to respond. When 268 people in the 22-29 age bracket are contacted, 242 respond and 26 refuse to respond. Suppose that one of the 349 people is randomly selected. Find the probability of getting someone in the 18–21 age bracket or someone who refused to respond.

Calculate the probability of selecting someone in the 18–21 age bracket or someone who refused to respond from a survey sample. This problem covers probability concepts, including the inclusion-exclusion principle, with clear step-by-step calculations. Suitable for students in Grades 9-12.

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