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#KB Probability Theory — Part 2- Common Discrete Probability Distributions

Dear Statisticians!

How often do you expect your favorite coffee shop to get swamped with customers all at once? Imagine it’s 8:30 AM on a typical weekday, and suddenly a rush of 15 people walks in. The baristas at Ella’s Cold Brew for Deadline Dashers scramble to serve everyone, but what if we could predict how many such spikes might happen, so they could prepare ahead? This is where understanding probability distributions, like the Poisson and Geometric distributions, becomes critical for managing customer flow, handling rare events, or even improving sales strategies.

When we first discussed discrete probability distributions, we focused on the binomial distribution — which calculates the likelihood of a specific number of successes in a fixed number of trials, like tracking the open rate of an email campaign. But often, businesses need to model events that happen over time or without a fixed number of attempts. For instance, you might want to know how many customer inquiries your support team will handle in a day or how many defects will show up on a production line. That’s where the Poisson distribution becomes useful. Similarly, if you’re analyzing how many follow-up emails a sales team must send before one results in a purchase, the geometric distribution is the right tool, as it models the probability of success after multiple unsuccessful attempts.

Poisson Distribution

The Poisson distribution (“pwah-sohn") is used to model the probability of a certain number of events occurring within a fixed interval of time or space. It’s especially useful for predicting rare events, such as customer arrivals or defects in manufacturing, when these events happen independently and at a constant average rate.

Key characteristics of the Poisson distribution include:

It deals with discrete, countable events.
Events occur independently of each other.
The average rate of occurrence, denoted by λ\lambdaλ, is constant.
Two or more events cannot occur at exactly the same instant.

In the context of Ella’s Cold Brew for Deadline Dashers, you might use the Poisson distribution to predict how many customers will arrive during the peak morning rush. Let’s say the average number of student customers coming in during the 8:30 to 9:30 AM hour is 15. If you want to know the probability of exactly 20 students arriving within that time, the Poisson formula is:

Where:

P (X = k) is the probability of seeing k customers (in this case, 20).
λ (lambda) is the average rate (15 students per hour).
e is Euler’s number (approximately 2.718).
k is the actual number of customers you’re interested in (20).

The Poisson distribution is useful for handling sudden spikes in customer numbers, so businesses like Ella’s can better plan staffing and inventory to match student traffic without wasting resources.

In the Poisson distribution, λ (lambda) represents the expected or average number of events occurring within a specified time period or space. For Ella’s coffee shop, λ = 15 means that, on average, 15 students are expected to arrive between 8:30 and 9:30 AM. Lambda reflects the long-term average rate of occurrences and is important because it remains constant across similar intervals, making predictions reliable over time. The larger the value of λ, the higher the probability of seeing more events during that period.

Business Applications of Poisson Distribution

The Poisson distribution has broad applications beyond customer arrivals. It’s useful in any situation where you need to predict the frequency of independent events occurring at a constant average rate over a given period. Here are a few other business use cases:

Website Traffic Spikes: Websites often see sudden increases in traffic during promotions or important events. Using the Poisson distribution, site managers can estimate the number of visits per hour. For example, if a site usually gets 100 visits per hour, λ=100 can help predict how likely it is to get 120 or more visits in a certain period. This helps them manage server resources better.
Number of Defects in a Production Batch: Manufacturers often use the Poisson distribution to predict how many defects might appear in a batch of products. For example, if a factory typically finds 3 defects per 1,000 units, with λ = 3, the distribution helps calculate the chance of finding 5 or more defects in a batch. This information supports quality control and resource management.
Number of Accidents at an Intersection: Traffic engineers use the Poisson distribution to estimate how many accidents might happen at a specific intersection over time. If an intersection has an average of 2 accidents per month (λ = 2), they can calculate the chance of 4 or more accidents in a month. This helps them plan for safety improvements and emergency response.

Excel Example: Calculating Poisson Probabilities

To calculate Poisson probabilities in Excel, you can use the POISSON.DIST function, which simplifies the process of determining the likelihood of a specific number of events happening within a fixed period.

Let’s say Ella’s Cold Brew for Deadline Dashers averages 15 students per hour between 8:30 and 9:30 AM, and you want to find the probability of exactly 20 students arriving during this time. You would use the following formula in Excel:

=POISSON.DIST(20, 15, FALSE)

Here’s how the inputs work:

20 represents the number of students you're interested in (the actual event count).
15 is the average number of students arriving per hour, which is your λ\lambdaλ value.
FALSE specifies that you want the probability of exactly 20 students, rather than a cumulative probability.

When you enter this formula into Excel, it returns a probability value of 0.0418, or 4.18%. This means that there is a 4.18% chance that exactly 20 students will show up during the peak hour from 8:30 to 9:30 AM. This relatively low probability highlights the fact that while 20 students could arrive, it’s more likely that the actual count will be closer to the average of 15.

Geometric Distribution

The geometric distribution represents the probability of getting the first success after several independent failures in a series of identical trials. Unlike the binomial distribution, which counts the number of successes in a set number of trials, the geometric distribution is concerned with when the first success occurs. The main assumption is that each trial has the same probability of success, and the trials are independent of one another.

In business, this distribution is useful when you’re interested in how many attempts are needed to achieve a particular outcome. For example, at Ella’s Cold Brew for Deadline Dashers, if you’re trying to analyze how many promotional emails must be sent before a student makes a purchase, the geometric distribution can help you model this process.

The probability that the first success occurs on the k-th trial is given by:

Where:

P (X = k) is the probability that the first success occurs on the k-th trial.
p is the probability of success on each individual trial (e.g., a student making a purchase after receiving an email).
k is the trial number when the first success occurs.

Business Applications of the Geometric Distribution

The geometric distribution is useful for modeling situations where you need to know how many attempts it will take to achieve the first success. Here are a few business applications:

Sales Calls: Many businesses use cold calls or follow-ups to gain new clients. The geometric distribution helps sales teams predict how many calls they might need to make before closing a deal. For example, if a salesperson has a 5% chance of success with each call, the distribution estimates the chances of closing the deal after a certain number of tries. This helps the team plan how persistent they should be and manage their resources.

Customer Service Escalations: In customer support, the geometric distribution can model how many escalations may be needed to resolve an issue. If a representative has a 20% chance of solving a problem with each escalation, the distribution helps estimate how many escalations might happen before the issue is resolved. This helps management plan support workloads more effectively.

Excel Example: Calculating Geometric Probabilities

Unlike the Poisson and binomial distributions, Excel doesn’t have a built-in function for the geometric distribution. However, we can easily calculate it using a combination of existing functions. For example, if Ella’s Cold Brew estimates that there’s a 10% chance that a promotional email will result in a purchase, you can calculate the probability that the first sale will happen after exactly 5 emails using this formula in Excel:

= (1 - 0.1)^(5 - 1) * 0.1

Here’s how the inputs work:

(1 - 0.1) represents the probability of failure on each email attempt (i.e., 90% chance of no purchase).
5 - 1 accounts for the four failed emails before the first success.
0.1 is the probability of success on the fifth email.

This means that there’s a 6.56% chance that the first purchase will occur after sending exactly 5 emails. This helps Ella’s marketing team set expectations and plan follow-up efforts, knowing how many emails they may need to send before a student makes a purchase.

What if Ella wants to know the chance of making a sale within 5 emails, instead of just on the 5th email? This is where cumulative probabilities help. To figure out the likelihood of getting the first sale after 5 or fewer emails, you add up the chances of success for each try from the 1st to the 5th.

The cumulative probability for success in up to 5 attempts is:

P(X ≤ 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

In Excel, you can calculate this cumulative probability more easily using a formula.

el, this cumulative probability can be calculated more efficiently with the formula:

= 1 - (1 - 0.1)^5

For Ella’s marketing team, this formula returns 40.95%, meaning there’s a good chance that a student will make a purchase within the first 5 emails sent.

Comparing Discrete Distributions

Selecting the right discrete probability distribution depends on the nature of the data and the specific problem you’re addressing:

Binomial Distribution: Use this when you’re counting the number of successes in a set number of independent trials. Each trial has two outcomes — success or failure — and the success chance stays the same each time. A typical example in business would be determining the likelihood of a certain number of email recipients opening an email in a marketing campaign.
Poisson Distribution: This distribution is appropriate when you’re counting the number of independent events occurring over a continuous period of time or space, with the events happening at a constant average rate. Use the Poisson distribution when there is no fixed number of trials. For example, it’s useful for estimating the number of students arriving at Ella’s Cold Brew during a particular time frame.
Geometric Distribution: The geometric distribution should be used when you want to know how many trials are needed before the first success. Unlike the binomial distribution, there is no fixed number of trials. Instead, you’re interested in the probability of achieving the first success after a series of failures. For example, this distribution is helpful when predicting how many emails or cold calls it takes before making a sale.

It’s also worth noting that these distributions can be interconnected. For example, as the number of trials n increases and the probability of success p decreases in a binomial distribution, it begins to resemble a Poisson distribution with λ = np. This is particularly useful when dealing with large datasets where modeling rare events becomes more efficient with the Poisson approximation.

Similarly, the geometric distribution is a special case of the negative binomial distribution, which models the number of successes before a specified number of failures occurs. While the geometric distribution focuses on achieving the first success, the negative binomial allows for more complex scenarios where multiple successes are needed, making it a flexible tool in many business applications.