To find the probability of having sampled 3 or fewer people that voted, we can calculate the probability of sampling 0, 1, 2, and 3 voters and then add those probabilities together.
First, we need to calculate the probability of sampling a voter who voted. Since 39% of eligible voters voted, the probability of sampling a voter who voted is 0.39.
Next, we will use the binomial probability formula to calculate the probability of each specific outcome:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k),
where n is the number of trials (7 in this case), k is the number of successful trials (the number of voters sampled that voted), p is the probability of success (0.39), and (n choose k) is the number of ways to choose k successes from n trials, calculated by the binomial coefficient formula:
(n choose k) = n! / (k! * (n-k)!).
Now, let's calculate the probabilities for each specific outcome:
P(X=0) = (7 choose 0) * 0.39^0 * (1-0.39)^(7-0) = 1 * 1 * 0.61^7 = 0.061154.
P(X=1) = (7 choose 1) * 0.39^1 * (1-0.39)^(7-1) = 7 * 0.39 * 0.61^6 = 0.202327.
P(X=2) = (7 choose 2) * 0.39^2 * (1-0.39)^(7-2) = 21 * 0.39^2 * 0.61^5 = 0.283986.
P(X=3) = (7 choose 3) * 0.39^3 * (1-0.39)^(7-3) = 35 * 0.39^3 * 0.61^4 ≈ 0.247368.
Finally, we can add these probabilities together to find the probability of sampling 3 or fewer voters:
P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.061154 + 0.202327 + 0.283986 + 0.247368 ≈ 0.794835.
Therefore, the probability of having sampled 3 or fewer people that voted is approximately 0.794835 or 79.48%.
In 2002, approximately 39% of the people eligible to vote in mid-term elections voted. Suppose you have taken a sample of 7 eligible voters. What is the probability of having sampled 3 or fewer people that voted?
1 answer
1 answer