A candidate for public office has claimed that 60% of voters will vote for her. If 5 registered voters were sampled,
1. What is the probability distribution function for this voting?
2. What is the probability that exactly 3 would say they favor this candidate?
3. Calculate the mean and standard deviation of the probability distribution
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To find the probability distribution function (PDF) for this voting scenario, we can use the binomial distribution formula.
1. Probability Distribution Function (PDF):
The binomial distribution formula is given by:
P(X = k) = C(n, k) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of exactly k successes (in this case, voters favoring the candidate).
- C(n, k) is the number of combinations of n items taken k at a time.
- p is the probability of success (voter favoring the candidate), which is given as 0.6.
- q is the probability of failure (voter not favoring the candidate), which is given as 0.4.
- n is the total number of trials or samples, which is 5 in this case.
Now, let's calculate the PDF for this scenario:
P(X = 0) = C(5, 0) * 0.6^0 * 0.4^5
P(X = 1) = C(5, 1) * 0.6^1 * 0.4^4
P(X = 2) = C(5, 2) * 0.6^2 * 0.4^3
P(X = 3) = C(5, 3) * 0.6^3 * 0.4^2
P(X = 4) = C(5, 4) * 0.6^4 * 0.4^1
P(X = 5) = C(5, 5) * 0.6^5 * 0.4^0
2. Probability that exactly 3 voters favor the candidate:
To find the probability that exactly 3 voters favor the candidate, we need to calculate P(X = 3) using the PDF formula.
P(X = 3) = C(5, 3) * 0.6^3 * 0.4^2
The value obtained will give us the probability of exactly 3 voters favoring the candidate.
3. Mean and Standard Deviation:
The mean (μ) and standard deviation (σ) of the probability distribution can be calculated using the following formulas:
Mean (μ) = n * p
Standard Deviation (σ) = sqrt(n * p * q)
Substituting the given values:
Mean (μ) = 5 * 0.6
Standard Deviation (σ) = sqrt(5 * 0.6 * 0.4)
Calculating these values will provide the average number of voters favoring the candidate and the measure of spread around the mean, respectively.
To understand the probability distribution in this scenario, we can assume that each registered voter has an independent probability of voting for the candidate. We can also assume that the claim of 60% support holds true for the entire population.
1. Probability Distribution Function (PDF):
The probability distribution function for this voting can be modeled using the binomial distribution, as we are interested in the number of successes (voters favoring the candidate) among a fixed number of trials (sampling of registered voters). The PDF of the binomial distribution is given by:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of exactly x successes.
- C(n, x) is the binomial coefficient, calculated as n! / (x! * (n-x)!).
- p is the probability of success (favoring the candidate).
- (1-p) is the probability of failure (not favoring the candidate).
- n is the number of trials (sample size).
- x is the number of successes (voters favoring the candidate).
2. Probability of exactly 3 voters favoring the candidate:
To find the probability that exactly 3 out of 5 voters favor the candidate, we can substitute the values into the PDF formula:
P(3) = C(5, 3) * 0.6^3 * (1-0.6)^(5-3)
Calculating this expression will give us the desired probability.
3. Mean and Standard Deviation:
For a binomial distribution, the mean (μ) and standard deviation (σ) can be calculated using the formulas:
μ = n * p
σ = sqrt(n * p * (1-p))
For this scenario, n = 5 (sample size) and p = 0.6 (probability of favoring the candidate).