Using Probability Distributions to Solve Business Problems

Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

Probability distributions can be useful tools for analyzing business issues, particularly those that involve risk or uncertainty. This lesson reviews three common probability distributions used to solve typical business problems.

How Good Are Your Predictions?

In business it's impossible to make predictions that are guaranteed to become reality. In trying to predict the probable outcomes of their decisions, many businesses rely on probability analysis to reduce uncertainty and make better-informed decisions.

When measured using many trials, probability can be viewed or expressed statistically as a probability distribution. There are many different distribution profiles; the binomial, hypergeometric, and Poisson distributions are often used in making business or economic decisions. In order to study how these work, let's go skiing!

A Ski Rental Shop

Actually, before we can spend time on the slopes we need to earn some money, which we are going to do by managing an equipment rental shop. One of the first issues we need to address is our inventory. For example, how confident are we that we have enough equipment to rent out, without having too much unused surplus?

In this case we can use a binomial distribution. As its name implies, a binomial distribution is based on two possible results. A typical example of a binomial distribution is tossing a coin: you either get heads or tails. Many business issues can also be expressed in binary terms. For example, if you advertise in the local media this week, do your sales go up the next week or not?

In order to use a binomial distribution, we need to know how many events we're measuring and what the probability of individual success or failure is. Most importantly, each individual result must be independent of the results from any other trial. Given this, a binomial distribution measures the probability of a specified number of positive or negative results.

In our rental shop case, we know that on the busiest day we can expect 150 rentals, which forms the number of independent events or trials. We also know that, historically, 60% of our customers rent skis and 40% rent snowboards, which provides our probability. If we decide that we only need to have 65 snowboards in stock, what is the probability that we will run out of snowboard rentals on any specific day?

Formulas and lookup tables can be used for these calculations, but it is common to use a spreadsheet or statistical program to calculate the binomial probability. In this case, we want to know the probability that 66 or more customers out of 150 will want to rent a snowboard.

P(failure>65, trials=150, probability=0.40) = 13.9%.

This number is statistically significant, and indicates that we should increase our stock. For example, simply increasing our stock to 70 results in a much lower chance of failure:

P(failure>70, trials=150, probability=0.40) = 2.8%.

Are Trials Independent or Linked?

Similar to the binomial distribution, a hypergeometric distribution also relies upon binary results. The key difference is that, unlike the binomial distribution, a hypergeometric distribution can be used when individual samples are not independent. This typically occurs when we do not replace items from the population as they are sampled.

An easy way to think about this comes from sampling a deck of cards. If you select a single card, you have a 25% probability that it will be one particular suit. If you replace the card and shuffle the deck, when you select another card you have the exact same probability. However, suppose you select a card, but then leave that card out of the deck and select again. In this case, you have reduced the probability of selecting that particular suit from the remaining cards. With relatively small sample sizes, this process becomes statistically significant.

Let's go back to our ski rental shop and imagine that we decide to run a promotion by randomly giving away snowboards. To do this, we bake and wrap 150 cookies, and in five of those we include a voucher for one free board. As each customer checks their equipment in, they are offered a free cookie. If 85 of our 150 customers actually take a cookie, what is the chance that we do not have to give away any boards? Because we're not replacing cookies as they are taken from the population, we use the hypergeometric distribution to calculate this result.

P(success=0, samples=85, total=5, pop=150) = 1.4%

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