A company has a call center for the purpose of responding to telephone inquiries

Question

A company has a call center for the purpose of responding to telephone inquiries from its customers. A telephone call received from this center is one of either two types: call from an existing customer or call from a new customer. For each hour, the number of calls received from this call center has a Poisson distribution with mean 10 and the percentage of calls from existing customers is 40%. (a) (5 points) Calculate the probability that on a given hour, there are more than 3 calls from a new customer. (b) (8 points) Calculate the conditional probability that if there is exactly one call made on a given hour, then it is a call made from an existing customer. (c) Suppose that this same company has 100 call centers all over the world. Each call center are independently run and receives calls from either existing or new customers in exactly the same fashion as described above. (i) (2 points) Describe the distribution of N, the total number of calls received from all 100 call centers on any given hour. (ii) (5 points) Use a suitable approximation to estimate the probability that on a given hour, there are at most 1,100 calls received from all call centers.

Explanation / Answer

When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities describe the statistical number of outcomes considered divided by the number of all outcomes (tossing a fair coin twice will yield HH with probability 1/4, because the four outcomes HH, HT, TH and TT are possible). When it comes to practical application, however, the word probability does not have a singular direct definition. In fact, there are two major categories of probability interpretations, whose adherents possess conflicting views about the fundamental nature of probability: Objectivists assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.[4] A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once. Subjectivists assign numbers per subjective probability, i.e., as a degree of belief.[5] The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.[6] Starting from arbitrary, subjective probabilities for a group of agents, some Bayesians[who?] claim that all agents will eventually have sufficiently similar assessments of probabilities, given enough evidence.

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