Understanding the Probability of All Operators Being Busy Simultaneously in a Help Desk

A particular help desk features 15 operators on duty, with an average of only 14 operators simultaneously busy. This raises an interesting question: what is the probability that all 15 operators will be busy at the same time?

Modeling the Situation with Binomial Distribution

The scenario can be modeled using a binomial distribution, where:

n 15, the total number of operators. p is the probability that an operator is busy.

Given that 14 operators are on average busy, we can express the average number of busy operators as:

Average busy operators n p

Solving for 'p'

By substituting the known values, we get:

14 15 p

Solving for p:

p frac{14}{15} approx 0.9333

Calculating the Probability Using Binomial Distribution

Next, we want to find the probability that all 15 operators are busy. This corresponds to the probability that X 15 in a binomial distribution where

X sim Binomial(n15, pfrac{14}{15})

The probability mass function of a binomial distribution is given by:

P(Xk) binom{n}{k} p^k (1-p)^{n-k}

Substituting n 15, k 15, and p frac{14}{15}, we get:

P(X 15) binom{15}{15} left(frac{14}{15}right)^{15} left(1 - frac{14}{15}right)^{0}

Calculating the Probability

After performing the calculation, we find:

left(frac{14}{15}right)^{15} approx 0.368

Therefore, the probability that all 15 operators will be simultaneously busy is approximately 0.368 or 36.8%.

Verification through Simulation

I simulated the situation on an Excel spreadsheet, where one operator was randomly left not busy, over one million trials. The result showed that 35.6 percent of the time, all 15 operators were simultaneously busy.

Factors Influencing the Probability

The probability of all operators being busy can be influenced by several factors, including:

The average call time. The inbound rate of calls. After-call documentation and record management.

Additional Observations

On average, fourteen operators represent about 93.33% of the total of fifteen operators. Therefore, adding an additional fifteenth operator would require 100% of the operators to be busy, making the probability 6.67%.

Dependence on factors like the average call time and inbound rate of calls is crucial. For two extremes, if always exactly 14 operators are busy, the probability is 0. On the other hand, in a scenario where 1/15 of the time no operators are busy, and 93.33% of the time, all the operators are busy, the probability becomes 0.9333…