Question

in a certain town, 45% of the voters support a school referendum up for a vote. sharon will ask people about why they support the referendum, but she needs to find people who support it. what is the probability that it takes 7 people to find the first person who supports the referendum? assume that after a person is selected, they are still considered for later selections and thus could be selected again.

Explanation:

Step1: Identify the distribution

This is a geometric distribution problem, where we want the probability that the first success (finding a supporter) occurs on the 7th trial. The probability of success \( p = 0.45 \), so the probability of failure \( q = 1 - p = 0.55 \).

Step2: Apply the geometric probability formula

The formula for the probability that the first success occurs on the \( k \)-th trial in a geometric distribution is \( P(X = k) = q^{k - 1} \times p \). Here, \( k = 7 \), so we calculate \( P(X = 7) = (0.55)^{7 - 1} \times 0.45 \).

Step3: Calculate the values

First, calculate \( (0.55)^6 \). \( 0.55^6 \approx 0.01861 \). Then multiply by \( 0.45 \): \( 0.01861 \times 0.45 \approx 0.00837 \)? Wait, no, wait, maybe I miscalculated. Wait, let's recalculate \( 0.55^6 \):

\( 0.55^2 = 0.3025 \)

\( 0.55^3 = 0.55 \times 0.3025 = 0.166375 \)

\( 0.55^4 = 0.55 \times 0.166375 = 0.09150625 \)

\( 0.55^5 = 0.55 \times 0.09150625 = 0.0503284375 \)

\( 0.55^6 = 0.55 \times 0.0503284375 = 0.027680640625 \)

Then multiply by \( 0.45 \): \( 0.027680640625 \times 0.45 \approx 0.01245628828125 \approx 0.0125 \)? Wait, but the options include 0.0872, 0.0277, 0.0125, 0.4500. Wait, maybe I made a mistake. Wait, no, the problem says "it takes 7 people to find the first person who supports", so that means the first 6 are non-supporters and the 7th is a supporter. So \( P = (0.55)^6 \times 0.45 \). Let's compute \( 0.55^6 \):

\( 0.55^6 = (0.55)^2 \times (0.55)^2 \times (0.55)^2 = 0.3025 \times 0.3025 \times 0.3025 \). Wait, no, 0.3025 0.3025 = 0.09150625, then 0.3025 = 0.027680640625, which matches the earlier calculation. Then 0.027680640625 * 0.45 = 0.01245628828125 ≈ 0.0125. So the answer should be 0.0125.

Answer:

0.0125