In a survey of U.S. adults with a sample size of 

20052005​, 

322322 

said Franklin Roosevelt was the best president since World War II. 

TwoTwo 

U.S. adults are selected at random from this sample without replacement. Complete parts​ (a) through​ (d).

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Part 1

​(a) Find the probability that both adults say Franklin Roosevelt was the best president since World War II.

The probability that both adults say Franklin Roosevelt was the best president since World War II is 

0.0260.026. 

​(Round to three decimal places as​ needed.)

Part 2

​(b) Find the probability that neither adult says Franklin Roosevelt was the best president since World War II.

The probability that neither adult says Franklin Roosevelt was the best president since World War II is 

0.70.7. 

​(Round to three decimal places as​ needed.)

### Given:

- Total sample size = \( 2005 \)
- Number of people who said Franklin Roosevelt was the best president = \( 322 \)
- Two adults are selected without replacement.

### Part (a) Find the probability that both adults say Franklin Roosevelt was the best president since World War II:

1. The probability of the first adult saying Franklin Roosevelt was the best president is:

   \[
   P(\text{first adult says Roosevelt}) = \frac{322}{2005}
   \]

2. After the first adult is selected, there are now \( 322 - 1 = 321 \) Roosevelt supporters left and a total of \( 2005 - 1 = 2004 \) adults left in the sample. The probability that the second adult also says Roosevelt is:

   \[
   P(\text{second adult says Roosevelt}) = \frac{321}{2004}
   \]

3. Therefore, the probability that both adults say Franklin Roosevelt was the best president is:

   \[
   P(\text{both say Roosevelt}) = \frac{322}{2005} \times \frac{321}{2004}
   \]

   \[
   P(\text{both say Roosevelt}) \approx 0.026
   \]

Thus, the probability is approximately **0.026**.

---

### Part (b) Find the probability that neither adult says Franklin Roosevelt was the best president since World War II:

1. The probability of the first adult **not** saying Franklin Roosevelt was the best president is:

   \[
   P(\text{first adult not Roosevelt}) = \frac{2005 - 322}{2005} = \frac{1683}{2005}
   \]

2. After selecting the first adult, there are \( 2005 - 1 = 2004 \) people left, and \( 1683 - 1 = 1682 \) adults left who do not support Roosevelt. The probability that the second adult also does not say Roosevelt is:

   \[
   P(\text{second adult not Roosevelt}) = \frac{1682}{2004}
   \]

3. Therefore, the probability that neither adult says Franklin Roosevelt was the best president is:

   \[
   P(\text{neither says Roosevelt}) = \frac{1683}{2005} \times \frac{1682}{2004}
   \]

   \[
   P(\text{neither says Roosevelt}) \approx 0.7
   \]

Thus, the probability is approximately **0.7**.

Would you like further clarification or details on any of these steps?

---

Here are 5 related questions to expand on this topic:

1. What is the probability that exactly one of the two selected adults says Franklin Roosevelt was the best president?
2. How does the probability change if the sample is much smaller, say 500 adults?
3. How would the probability change if the selection was with replacement?
4. What is the expected number of Roosevelt supporters in a randomly selected group of 10 people?
5. Can we use a binomial distribution approximation for this problem? Why or why not?

**Tip:** When selecting items without replacement, the probabilities change after each selection. This is different from cases with replacement, where probabilities remain constant.

In a survey of U.S. adults with a sample size of 2005, 322 said Franklin Roosevelt was the best president since World War II. Two U.S. adults are selected at random from this sample without replacement. Complete parts (a) through (d). Part 1 (a) Find the probability that both adults say Franklin Roosevelt was the best president since World War II. The probability that both adults say Franklin Roosevelt was the best president since World War II is 0.026. (Round to three decimal places as needed.) Part 2 (b) Find the probability that neither adult says Franklin Roosevelt was the best president since World War II. The probability that neither adult says Franklin Roosevelt was the best president since World War II is 0.7. (Round to three decimal places as needed.)

This problem explores probability by calculating the chances that two randomly selected U.S. adults believe Franklin Roosevelt was the best president since World War II, with a sample size of 2005. Learn how to solve using key probability concepts and formulas. Suitable for Grades 10-12.

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