The question provided from the image involves probability calculations based on a survey of U.S. adults. The sample size is 2025, of which 333 adults said Franklin Roosevelt was the best president since World War II. Two U.S. adults are selected at random from the sample without replacement, and the problem requires solving four sub-problems related to this data.

Let's break down the individual parts:

1. Total population: $N = 2025$
2. Number of favorable cases (Roosevelt supporters): $F = 333$
3. Number of non-supporters: $N - F = 2025 - 333 = 1692$

The following are the individual sub-questions:

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

This is a "without replacement" problem, so we use conditional probability:

$P(\text{both}) = \frac{F}{N} \times \frac{F - 1}{N - 1}$

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

The probability that neither of the adults says Roosevelt was the best president is:

$P(\text{neither}) = \frac{N - F}{N} \times \frac{N - F - 1}{N - 1}$

(c) Find the probability that at least one of the two adults says Franklin Roosevelt was the best president since World War II.

This is the complement of both not supporting Roosevelt, so:

$P(\text{at least one}) = 1 - P(\text{neither})$

(d) Find the probability that exactly one of the two adults says Franklin Roosevelt was the best president since World War II.

This occurs in two cases: first adult says yes and second says no, or first says no and second says yes:

$P(\text{exactly one}) = \left(\frac{F}{N} \times \frac{N - F}{N - 1}\right) + \left(\frac{N - F}{N} \times \frac{F}{N - 1}\right)$

Now I will compute these values.Here are the probabilities based on the calculations:

• (a) The probability that both adults say Franklin Roosevelt was the best president since World War II is 0.027.
• (b) The probability that neither adult says Franklin Roosevelt was the best president since World War II is 0.698.
• (c) The probability that at least one of the two adults says Franklin Roosevelt was the best president since World War II is 0.302.
• (d) The probability that exactly one of the two adults says Franklin Roosevelt was the best president since World War II is 0.275.

Do you want any further clarification or details on these calculations?

Here are 5 related questions to deepen understanding:

1. What is the difference between "without replacement" and "with replacement" in probability problems?
2. How would the probabilities change if the sample size were much larger?
3. Can you explain how conditional probability applies in this problem?
4. How would the problem be different if we selected three adults instead of two?
5. What is the significance of complementary probabilities in this scenario?

Tip: Always check whether the problem specifies "with or without replacement," as it greatly affects the probability calculations.