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Question Survey of 2017 U.S. adults, 383 said FDR was best president since WWII. Find probability that 2 randomly selected adults both say FDR was best. P(both say FDR) = 3832017⋅3822016\frac{383}{2017} \cdot \frac{382}{2016}.

Studdy Solution

STEP 1

Assumptions
1. The total sample size is 2017 U.S. adults.
2. Out of the sample, 383 adults say Franklin Roosevelt was the best president since World War II.
3. Two U.S. adults are selected at random without replacement.
4. We need to find the probability that both selected adults say Franklin Roosevelt was the best president since World War II.

STEP 2

To find the probability that both adults selected say Franklin Roosevelt was the best president since World War II, we need to calculate the probability of selecting one adult who thinks this way and then another, without replacement.
P(both say FDR is best)=P(first says FDR is best)×P(second says FDR is best | first said FDR is best)P(\text{both say FDR is best}) = P(\text{first says FDR is best}) \times P(\text{second says FDR is best | first said FDR is best})

STEP 3

First, calculate the probability that the first adult selected says Franklin Roosevelt was the best president since World War II.
P(first says FDR is best)=number who say FDR is besttotal sample sizeP(\text{first says FDR is best}) = \frac{\text{number who say FDR is best}}{\text{total sample size}}

STEP 4

Plug in the values for the number who say FDR is best and the total sample size.
P(first says FDR is best)=3832017P(\text{first says FDR is best}) = \frac{383}{2017}

STEP 5

Next, calculate the probability that the second adult selected says Franklin Roosevelt was the best president since World War II, given that the first one already did. Since one person who says FDR is the best has been removed from the sample, both the numerator and the denominator decrease by 1.
P(second says FDR is best | first said FDR is best)=number who say FDR is best−1total sample size−1P(\text{second says FDR is best | first said FDR is best}) = \frac{\text{number who say FDR is best} - 1}{\text{total sample size} - 1}

STEP 6

Plug in the values for the number who say FDR is best and the total sample size, both decreased by 1.
P(second says FDR is best | first said FDR is best)=383−12017−1P(\text{second says FDR is best | first said FDR is best}) = \frac{383 - 1}{2017 - 1}

STEP 7

Simplify the expression for the probability that the second adult selected says Franklin Roosevelt was the best president since World War II.
P(second says FDR is best | first said FDR is best)=3822016P(\text{second says FDR is best | first said FDR is best}) = \frac{382}{2016}

STEP 8

Now, multiply the probabilities from STEP_4 and STEP_7 to find the probability that both adults say Franklin Roosevelt was the best president since World War II.
P(both say FDR is best)=3832017×3822016P(\text{both say FDR is best}) = \frac{383}{2017} \times \frac{382}{2016}

STEP 9

Calculate the probability by multiplying the fractions.
P(both say FDR is best)=383×3822017×2016P(\text{both say FDR is best}) = \frac{383 \times 382}{2017 \times 2016}

STEP 10

Perform the multiplication in the numerator and the denominator.
P(both say FDR is best)=1463664068272P(\text{both say FDR is best}) = \frac{146366}{4068272}

STEP 11

Divide the numerator by the denominator to get the probability in decimal form.
P(both say FDR is best)=1463664068272≈0.03597P(\text{both say FDR is best}) = \frac{146366}{4068272} \approx 0.03597

STEP 12

Round the probability to three decimal places as required.
P(both say FDR is best)≈0.036P(\text{both say FDR is best}) \approx 0.036
The probability that both adults say Franklin Roosevelt was the best president since World War II is 0.0360.036.

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