A telemarketer with B2B Communications has a 20 percent historical probability of making a sale during a shift where 100 calls were made. Assign a random variable x, where the value of x is equal to the number of sales made during the shift. Describe the probability distribution of x, and calculate the probability of the employee making exactly 28 sales during a 100-call shift. What is the probability that the employee will make less than 28 sales during the 100-call shift? What is the probability that the employee will make 28 or more sales during the 100-call shift?

Question

A telemarketer with B2B Communications has a 20 percent historical probability of making a sale during a shift where 100 calls were made. Assign a random variable x, where the value of x is equal to the number of sales made during the shift. Describe the probability distribution of x, and calculate the probability of the employee making exactly 28 sales during a 100-call shift. What is the probability that the employee will make less than 28 sales during the 100-call shift? What is the probability that the employee will make 28 or more sales during the 100-call shift?

Answer 1

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Answer 2

To describe the probability distribution of the random variable x, we can use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes, success or failure, and each outcome has a fixed probability of occurring.

In this case, the success is making a sale, and the failure is not making a sale. The probability of success is 20% or 0.2, which means the probability of failure is 1 - 0.2 = 0.8.

The random variable x represents the number of sales made during a 100-call shift, so x can take values from 0 to 100.

The probability of making exactly 28 sales during a 100-call shift can be calculated using the binomial distribution formula:

P(x = 28) = (100 choose 28) * (0.2^28) * (0.8^(100-28))

To calculate this using a calculator, you can use the combination function (nCr) to compute (100 choose 28), and then multiply it by (0.2^28) and (0.8^(100-28)).

The probability that the employee will make less than 28 sales during the 100-call shift can be calculated by summing up the probabilities of making 0, 1, 2, ..., 27 sales:

P(x < 28) = P(x = 0) + P(x = 1) + ... + P(x = 27)

Similarly, the probability that the employee will make 28 or more sales during the 100-call shift can be calculated by summing up the probabilities of making 28, 29, ..., 100 sales:

P(x >= 28) = P(x = 28) + P(x = 29) + ... + P(x = 100)

You can calculate these probabilities using a calculator or statistical software, or you can use a binomial distribution table to look up the probabilities.