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 3 people that voted?
To calculate the probability of exactly 3 out of 7 eligible voters voting, we use the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials or eligible voters sampled (7)
- k is the number of successful outcomes or voters that voted (3)
- (n choose k) is the number of ways to choose k successes out of n trials (calculated as n! / (k! * (n-k)!))
- p is the probability of a successful outcome or the proportion of eligible voters that vote (0.39)
- (1-p) is the probability of a failure (1-0.39)
Plugging in the values:
P(X=3) = (7 choose 3) * 0.39^3 * (1-0.39)^(7-3)
Calculating the binomial coefficient:
(7 choose 3) = 7! / (3! * (7-3)!)
= 7! / (3! * 4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
Calculating the probability:
P(X=3) = 35 * 0.39^3 * (1-0.39)^(7-3)
= 35 * 0.39^3 * 0.61^4
≈ 0.2812
Therefore, the probability of having sampled exactly 3 people that voted is approximately 0.2812, or 28.12%.
To calculate the probability of sampling exactly 3 people who voted, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials (in this case, the number of voters sampled)
- k is the number of successes (in this case, the number of people who voted)
- (n C k) is the binomial coefficient, which represents the number of possible combinations of k successes out of n trials
- p is the probability of a success (in this case, the probability of an eligible voter voting)
- (1 - p) is the probability of a failure (in this case, the probability of an eligible voter not voting)
- ^ represents exponentiation
In this scenario:
- n = 7 (the number of voters sampled)
- k = 3 (the number of people who voted)
- p = 0.39 (the probability of an eligible voter voting)
Using these values, we can calculate the probability as follows:
P(X = 3) = (7 C 3) * 0.39^3 * (1 - 0.39)^(7 - 3)
First, let's calculate the binomial coefficient:
(7 C 3) = (7! / (3!(7 - 3)!)) = (7! / (3! * 4!)) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Now, let's calculate the probability:
P(X = 3) = 35 * 0.39^3 * (1 - 0.39)^(7 - 3)
P(X = 3) = 35 * 0.0391 * 0.61^4
P(X = 3) = 0.3868
Therefore, the probability of having sampled exactly 3 people who voted is approximately 0.3868, or 38.68%.
To find the probability of sampling exactly 3 people who voted, we can use the binomial probability formula.
The binomial probability formula states that the probability of having exactly "k" successes in "n" trials, where the probability of success in each trial is "p," is given by the formula:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes
- C(n,k) is the combination formula, also known as "n choose k," and it represents the number of ways to choose k items from a set of n items
- p is the probability of success in each trial
- (1-p) is the probability of failure in each trial
- n is the number of trials
In this case, the probability of an eligible voter voting is 39%, which can be written as 0.39. So p = 0.39.
Since we have taken a sample of 7 eligible voters, n = 7.
We want to find the probability of exactly 3 people voting, so k = 3.
Now we can plug these values into the formula:
P(X=3) = C(7,3) * 0.39^3 * (1-0.39)^(7-3)
C(7,3) = 7! / (3!(7-3)!) = 35
P(X=3) = 35 * 0.39^3 * 0.61^4
P(X=3) = 0.3651
Therefore, the probability of having sampled exactly 3 people who voted is approximately 0.3651.