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. A binomial distribution is used when there are two possible outcomes (in this case, success or failure) and each trial is independent.
In this case, we know that Susan has a success rate of 67% or 0.67. Let's calculate the probabilities step by step:
A.) Compute P(n=12), P(n=13), and P(n=14):
To calculate P(n=k), we can use the formula for the binomial distribution:
P(n=k) = C(n-1, k-1) * p^k * (1-p)^(n-k)
where
C(n, k) is the combination formula (n choose k),
p is the probability of success, and
n is the number of trials.
Using this formula, we can calculate the probabilities:
P(n=12) = C(11, 11) * (0.67)^12 * (1-0.67)^(12-12)
P(n=13) = C(12, 11) * (0.67)^13 * (1-0.67)^(13-11)
P(n=14) = C(13, 11) * (0.67)^14 * (1-0.67)^(14-11)
By substituting the values into the formula and evaluating the combinations (C), we can find the probabilities.
B.) To find the probability that Susan will need from 12 to 14 contacts to get the bonus, we need to calculate P(n=12) + P(n=13) + P(n=14).
C.) To find the probability that Susan will need more than 14 contacts to get the bonus, we need to calculate 1 - (P(n=12) + P(n=13) + P(n=14)).
D.) To calculate the expected value (μ) and standard deviation (σ) of the random variable n, we can use the following formulas:
μ = n * p
σ = sqrt(n * p * (1-p))
where p is the probability of success, and n is the number of trials.
By substituting the values of p and calculating the expected value and standard deviation using these formulas, we can find the results.
Remember to use a calculator or software that can perform binomial distribution calculations easily if needed.