45.4% of people think the economy is the most important issue, 20.9% think it's the federal deficit, and the rest think it's something else. If 5 people are chosen at random, what is that probability that at least one will say the federal deficit is the most important issue?

To solve this problem, we need to find the probability that at least one of the five people chosen will say the federal deficit is the most important issue. Here's a step-by-step breakdown:

### Step 1: Define the probabilities
- Probability that a person thinks the federal deficit is the most important issue: \( P(\text{deficit}) = 0.209 \).
- Probability that a person does not think the federal deficit is the most important issue: 
  \[
  P(\text{not deficit}) = 1 - P(\text{deficit}) = 1 - 0.209 = 0.791.
  \]

### Step 2: Probability that none of the 5 people say the federal deficit
We need to find the probability that all 5 people chosen think the federal deficit is not the most important issue. This can be calculated as:

\[
P(\text{none say deficit}) = (P(\text{not deficit}))^5 = (0.791)^5.
\]

### Step 3: Calculate the probability that at least one person says the federal deficit
The probability that at least one person thinks the federal deficit is the most important issue is the complement of the probability that none of them think so:

\[
P(\text{at least one says deficit}) = 1 - P(\text{none say deficit}) = 1 - (0.791)^5.
\]

Let's calculate this value: 

\[
P(\text{at least one says deficit}) = 1 - 0.791^5.
\] 

Let's compute this value:The probability that at least one of the five people chosen will say the federal deficit is the most important issue is approximately **0.6903**, or 69.03%.

### Would you like more details or have any questions?

Here are some related questions to explore further:
1. What is the probability that exactly two people out of five will say the federal deficit is the most important issue?
2. How would the probability change if the sample size increased to 10 people?
3. What is the probability that none of the five people will think the economy is the most important issue?
4. If the probability of thinking the federal deficit is the most important issue changes to 25%, how does that affect the result?
5. How can this type of probability be used in real-world decision-making processes?

**Tip:** Remember, the complement rule is a powerful tool in probability calculations, especially when it's easier to find the probability of the opposite event.

Explore how to calculate the probability that at least one out of five randomly chosen people believes the federal deficit is the most important issue. This problem covers probability concepts, including the complement rule, and is suitable for Grades 9-12.

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