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 exactly 3 people that voted?
To find the probability of exactly 3 people out of 7 having voted, we need to use the binomial probability formula. The formula is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes (in this case, 3 people voted).
n is the total number of trials (sample size, which is 7 voters in this case).
k is the number of successful outcomes (people who voted, which is 3 in this case).
p is the probability of success in a single trial (39% or 0.39 since 39% of people voted).
(1 - p) is the probability of failure in a single trial (1 - 0.39 = 0.61).
Using this formula, we can calculate the probability:
P(X = 3) = (7 choose 3) * 0.39^3 * 0.61^(7 - 3)
First, let's calculate the combination (n choose k):
(7 choose 3) = 7! / (3! * (7 - 3)!) = (7! / (3! * 4!)) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Next, we plug these values into the formula:
P(X = 3) = 35 * 0.39^3 * 0.61^(7 - 3)
= 35 * 0.39^3 * 0.61^4 ≈ 0.2329
Therefore, the probability of exactly 3 out of 7 people having voted is approximately 0.2329, or 23.29%.
To calculate the probability of having exactly 3 people that voted, we can use the binomial probability formula.
The binomial probability formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k),
where:
P(X=k) is the probability of having exactly k successes,
C(n,k) is the number of combinations of n items taken k at a time,
p is the probability of success in a single trial,
n is the number of trials.
In this case, n = 7 (the sample size) and p = 0.39 (the probability of an eligible voter voting).
Now, let's substitute the values into the formula:
P(X=3) = C(7,3) * (0.39)^3 * (1-0.39)^(7-3).
To calculate C(7,3), we can use the formula:
C(n,k) = n! / (k! * (n-k)!),
where n! represents the factorial of n.
C(7,3) = 7! / (3! * (7-3)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.
Now, let's substitute all the values into the binomial probability formula:
P(X=3) = 35 * (0.39)^3 * (1-0.39)^(7-3).
P(X=3) = 35 * 0.39^3 * 0.61^4.
Calculating this expression will give you the probability of exactly 3 people voting.
To calculate the probability of exactly 3 people voting out of a sample of 7 eligible voters, we first need to determine the probability of an individual voter either voting or not voting.
In 2002, approximately 39% of eligible voters voted, which means the probability of an individual voter voting is 0.39, and the probability of an individual voter not voting is 0.61 (1 - 0.39).
To calculate the probability of exactly 3 people voting, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) represents the probability of exactly k successes.
- n represents the total number of trials (sample size).
- k represents the desired number of successes (in this case, 3).
- (n C k) represents the binomial coefficient, which is calculated by n! / (k! * (n - k)!), where ! denotes factorial.
- p represents the probability of success (individual voter voting).
- (1 - p) represents the probability of failure (individual voter not voting).
Substituting the values into the formula:
P(X = 3) = (7 C 3) * (0.39)^3 * (0.61)^(7-3)
To calculate the binomial coefficient (7 C 3):
(7 C 3) = 7! / (3! * (7 - 3)!)
= 7! / (3! * 4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
Substituting this value and the probabilities into the formula:
P(X = 3) = (35) * (0.39)^3 * (0.61)^(7-3)
= 35 * 0.39^3 * 0.61^4
≈ 0.2515
Therefore, the probability of having exactly 3 people out of a sample of 7 eligible voters who voted is approximately 0.2515, or 25.15%.