Questions: In a survey of U.S. adults with a sample size of 2083, 349 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). (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 (Round to three decimal places as needed.)

In a survey of U.S. adults with a sample size of 2083, 349 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).
(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 (Round to three decimal places as needed.)

Transcript text: In a survey of U.S. adults with a sample size of 2083, 349 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). (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 (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Probability that both selected items have the desired attribute

To calculate this, we first find the probability of the first item having the desired attribute, which is $\frac{K}{N} = \frac{349}{2083}$. Since the selection is without replacement, the probability for the second item having the desired attribute is $\frac{K-1}{N-1} = \frac{348}{2082}$. The combined probability is the product of these two probabilities: $\frac{K}{N} \times \frac{K-1}{N-1} = 0.028$.

Step 2: Probability that neither selected item has the desired attribute

The probability of the first selected item not having the desired attribute is $\frac{N-K}{N} = \frac{1734}{2083}$. For the second item, the probability is $\frac{N-K-1}{N-1} = \frac{1733}{2082}$. The combined probability is the product of these two probabilities: $\frac{N-K}{N} \times \frac{N-K-1}{N-1} = 0.693$.

Step 3: Probability that at least one of the selected items has the desired attribute

This can be found by subtracting the probability that neither selected item has the desired attribute from 1: $1 - \left(\frac{N-K}{N} \times \frac{N-K-1}{N-1}\right) = 0.307$.