Susan is a sales representative who has a history of making a successful sale from about 67% of her sales contacts. If she makes 12 successful sales this week, Susan will get a bonus. Let n be a random variable representing the number of contacts needed for Susan to get the 12th sale.
A.) Compute P(n=12), P(n=13), and P(n=14).
B.) What is the probability that Susan will need from 12 to 14 contacts to get the bonus?
C.) What is the probability that Susan will need more than 14 contacts to get the bonus?
D.) What are the expected value μ and standard deviation σ of the random variable n?
To answer these questions, we need to use the concept of a binomial distribution since Susan has a fixed probability of success (67%) for each contact.
The formula for the probability mass function (PMF) of a binomial distribution is given by:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the total number of trials (contacts)
- k is the number of successful trials (sales)
- p is the probability of success on each trial
Now let's answer each question:
A.) Compute P(n=12), P(n=13), and P(n=14):
To calculate these probabilities, we need to substitute the values into the binomial distribution formula. In this case, n=12 and p=0.67.
P(n=12) = (12 choose 12) * (0.67)^12 * (1-0.67)^(12-12)
P(n=13) = (13 choose 12) * (0.67)^12 * (1-0.67)^(13-12)
P(n=14) = (14 choose 12) * (0.67)^12 * (1-0.67)^(14-12)
You can use a calculator or a statistical software to calculate these probabilities.
B.) What is the probability that Susan will need from 12 to 14 contacts to get the bonus?
To find this probability, we can sum the individual probabilities calculated in part A.
P(12 ≤ n ≤ 14) = P(n=12) + P(n=13) + P(n=14)
C.) What is the probability that Susan will need more than 14 contacts to get the bonus?
To find this probability, we need to calculate the complement of the probability of getting the bonus within 14 contacts.
P(n > 14) = 1 - P(12 ≤ n ≤ 14)
D.) What are the expected value μ and standard deviation σ of the random variable n?
The expected value (mean) of a binomial distribution is given by μ = n * p, and the standard deviation is given by σ = sqrt(n * p * (1-p)).
Substitute the given values into the formulas to calculate μ and σ.
μ = n * p
σ = sqrt(n * p * (1-p))
These calculations will give us the expected value and standard deviation of the random variable n.
Note: The values of n, p, and the calculations will depend on the specific numbers provided in the question.
A.) To compute the probabilities P(n=12), P(n=13), and P(n=14), we need to use the binomial distribution formula:
P(n=k) = (nCk) * p^k * (1-p)^(n-k)
Where nCk is the binomial coefficient (number of ways to choose k items from a set of size n), p is the probability of success (successful sale rate), and k is the number of successful sales.
Given that Susan has a successful sale rate of 67%, we can calculate the probabilities as follows:
P(n=12) = (12C12) * (0.67)^12 * (1-0.67)^(12-12)
P(n=13) = (13C12) * (0.67)^12 * (1-0.67)^(13-12)
P(n=14) = (14C12) * (0.67)^12 * (1-0.67)^(14-12)
Calculating the values:
P(n=12) = (1) * (0.67)^12 * (0.33)^0
P(n=13) = (13) * (0.67)^12 * (0.33)^1
P(n=14) = (91) * (0.67)^12 * (0.33)^2
B.) To calculate the probability that Susan will need from 12 to 14 contacts to get the bonus, we sum the individual probabilities of P(n=12), P(n=13), and P(n=14):
P(12 ≤ n ≤ 14) = P(n=12) + P(n=13) + P(n=14)
C.) To calculate the probability that Susan will need more than 14 contacts to get the bonus, we need to sum the probabilities of all n-values greater than 14:
P(n > 14) = 1 - (P(n=12) + P(n=13) + P(n=14))
D.) To calculate the expected value μ (mean) and standard deviation σ (standard deviation) of the random variable n, we can use the following formulas:
μ = n * p
σ = sqrt(n * p * (1-p))
Where n is the number of trials and p is the probability of success.
In this case, since we want the expected value and standard deviation for the random variable n, we use the successful sale rate of Susan (67%) as the probability.
μ = n * p = n * 0.67
σ = sqrt(n * p * (1-p)) = sqrt(n * 0.67 * (1-0.67))
Therefore, we cannot calculate the exact values of μ and σ without knowing the value of n.