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?
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%.
To find the probability of having sampled 3 or fewer people that voted, we can use the binomial probability formula.
The formula for the probability of a binomial distribution is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of exactly k successes (in this case, the probability of having sampled k people that voted)
n is the total number of trials (in this case, the number of eligible voters sampled)
k is the number of desired successes (in this case, 3 or fewer people that voted)
p is the probability of success (in this case, the percentage of people that voted, which can be converted to a decimal)
Given that the percentage of people that voted is 39%, we can convert this to a decimal by dividing by 100:
p = 39% / 100 = 0.39
We want to find the probability of having sampled 3 or fewer people that voted, so we need to calculate the probability for each value of k from 0 to 3, and then sum those probabilities together.
P(X = 0) = (7 C 0) * (0.39)^0 * (1 - 0.39)^(7 - 0)
P(X = 1) = (7 C 1) * (0.39)^1 * (1 - 0.39)^(7 - 1)
P(X = 2) = (7 C 2) * (0.39)^2 * (1 - 0.39)^(7 - 2)
P(X = 3) = (7 C 3) * (0.39)^3 * (1 - 0.39)^(7 - 3)
To calculate the probabilities, we need to use the binomial coefficient (n C k), which represents the number of ways to choose k successes from n trials. The binomial coefficient can be calculated using the formula:
(n C k) = n! / (k! * (n - k)!)
Calculating each term and summing them together will give us the probability of having sampled 3 or fewer people that voted.
To find the probability of having sampled 3 or fewer people that voted, we first need to determine the probability of sampling exactly 3 people that voted, then the probability of sampling exactly 2 people that voted, then the probability of sampling exactly 1 person that voted, and finally the probability of sampling exactly 0 people that voted.
The probability of sampling exactly 3 people that voted can be calculated using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials (sample size)
- k is the number of successful outcomes (people that voted)
- (nCk) denotes the combination of n choose k
- p is the probability of a successful outcome (39% or 0.39 in decimal form)
- (1 - p) is the probability of an unsuccessful outcome
Using this formula, we can calculate the probability of having sampled 3 or fewer people that voted.