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 exactly 2 people that voted?
To calculate the probability of exactly 2 people voting out of a sample of 7, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (sample size)
- k is the number of successes (number of people voting)
- p is the probability of success (proportion of people voting)
In this case, n = 7, k = 2, and p = 0.39.
First, let's calculate the binomial probability:
P(X = 2) = C(7, 2) * (0.39)^2 * (1-0.39)^(7-2)
The combination function C(7, 2) can be calculated as:
C(7, 2) = 7! / (2! * (7-2)!)
= 7! / (2! * 5!)
= (7 * 6 * 5!)/ (2! * 5!)
= (7 * 6) / (2 * 1)
= 21
Now, we can substitute these values into the probability formula:
P(X = 2) = 21 * (0.39)^2 * (1-0.39)^(7-2)
P(X = 2) = 21 * 0.39^2 * 0.61^5
P(X = 2) ≈ 0.3196
Therefore, the probability of exactly 2 people voting out of a sample of 7 is approximately 0.3196, or 31.96%.
To calculate the probability of having sampled exactly 2 people who voted, we can use the binomial probability formula:
P(X=k) = C(n, k) * p^k * q^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes
- n is the total number of trials (in this case, the number of eligible voters sampled)
- k is the number of successes (in this case, the number of people who voted)
- C(n, k) is the combination formula, also known as the binomial coefficient, which calculates the number of ways to choose k successes from n trials
- p is the probability of success in a single trial (in this case, the probability of an eligible voter voting)
- q is the probability of failure in a single trial (in this case, the probability of an eligible voter not voting)
In this case:
- n = 7 (7 eligible voters were sampled)
- k = 2 (we want to find the probability of having exactly 2 people who voted)
- p = 0.39 (the probability of an eligible voter voting, which is 39% or 0.39)
- q = 1 - p = 1 - 0.39 = 0.61 (the probability of an eligible voter not voting)
Plugging in these values,
P(X=2) = C(7, 2) * 0.39^2 * 0.61^(7-2)
Now let's calculate it step by step.
C(7, 2) = 7! / (2!(7-2)!)
= 7! / (2! * 5!)
= (7 * 6 * 5!) / (2! * 5!)
= (7 * 6) / (2 * 1)
= 42 / 2
= 21
P(X=2) = 21 * 0.39^2 * 0.61^(7-2)
= 21 * 0.39^2 * 0.61^5
Calculating further,
P(X=2) ≈ 21 * 0.15129 * 0.137293
≈ 0.454078
Therefore, the probability of having sampled exactly 2 people who voted is approximately 0.454078, or 45.41%.
To calculate the probability of having sampled exactly 2 people who voted, we can use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (in this case, the sample size)
x is the number of successes (the number of people who voted)
p is the probability of success (the proportion of people who voted)
Now, let's plug in the values into the formula:
n = 7 (the sample size)
x = 2 (the number of people who voted)
p = 0.39 (the proportion of people who voted)
P(2) = (7C2) * (0.39)^2 * (1-0.39)^(7-2)
To calculate (7C2), or the number of ways to choose 2 people out of 7, we use the combination formula:
(7C2) = 7! / (2!(7-2)!)
= 7! / (2!5!)
= (7 * 6 * 5!) / (2! * 5!)
= 7 * 6 / 2!
= 21
Now, let's plug in the remaining values:
P(2) = 21 * (0.39)^2 * (1-0.39)^(7-2)
Calculating the rest of the expression:
P(2) = 21 * (0.39)^2 * (0.61)^5
≈ 0.2284
Therefore, the probability of having sampled exactly 2 people who voted is approximately 0.2284, or 22.84%.