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 0 people that voted?
The probability of having sampled exactly 0 people that voted is calculated using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (sample size)
- k is the number of successful outcomes (0 people voted in this case)
- p is the probability of a successful outcome (percentage of people who voted)
In this case, n = 7, k = 0, and p = 0.39.
P(X = 0) = (7 choose 0) * 0.39^0 * (1-0.39)^(7-0)
= 1 * 1 * 0.61^7
= 0.61^7
= 0.04131
Therefore, the probability of having sampled exactly 0 people that voted is approximately 0.04131 or 4.131%.
To find the probability of having sampled exactly 0 people that voted, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes,
n is the number of trials or sample size,
k is the number of successes,
p is the probability of success for each trial, and
(1-p) is the probability of failure for each trial.
In this case, n (sample size) is 7 and p (probability of success) is 0.39.
Let's plug in the values into the formula:
P(X = 0) = (7C0) * (0.39^0) * (1 - 0.39)^(7-0)
Calculating further:
(7C0) = 1 (as 7C0 is the number of ways to choose 0 successes out of 7)
(0.39^0) = 1 (any number raised to the power of 0 is equal to 1)
(1 - 0.39)^(7-0) = 0.61^7 = 0.15016151
Now we can substitute the values:
P(X = 0) = 1 * 1 * 0.15016151
The probability of having sampled exactly 0 people that voted is approximately 0.15016151 or 15.02%.
To calculate the probability of having sampled exactly 0 people that voted, you need to use the binomial probability formula. The binomial probability formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of having exactly k successes
- (n C k) is the number of combinations
- p is the probability of success on a single trial
- n is the total number of trials
- k is the number of desired successes
In this case, you want to find the probability of having 0 people vote, so k = 0. The probability of success (voting) is given as 39%, which can be expressed as 0.39. The total number of eligible voters sampled is n = 7.
Now, let's substitute these values into the formula:
P(X = 0) = (7 C 0) * (0.39^0) * (1 - 0.39)^(7 - 0)
To calculate (7 C 0), which represents the number of combinations, we use the combination formula:
(7 C 0) = 7! / (0! * (7 - 0)!) = 7! / (0! * 7!)
Since 0! is equal to 1, we can simplify further:
(7 C 0) = 7! / (1 * 7!) = 1
Now, let's continue calculating the probability:
P(X = 0) = 1 * (0.39^0) * (1 - 0.39)^7
Since any number raised to the power of 0 is 1, we simplify further:
P(X = 0) = 1 * 1 * (1 - 0.39)^7
P(X = 0) = (1 - 0.39)^7
Calculating this, we get:
P(X = 0) = 0.3522
Therefore, the probability of having sampled exactly 0 people that voted is approximately 0.3522.