Poisson Distribution: Definition, Formula & Examples

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  • 0:33 Poisson Distributions
  • 1:12 Probability Conditions
  • 1:44 Example One
  • 2:30 Example Two
  • 3:09 Example Three
  • 3:46 Lesson Summary
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Lesson Transcript
Instructor: Ryan Hultzman
In this lesson, we'll use a hypothetical road trip and some other real-life examples to show you how to use the Poisson distribution, a formula for calculating the probability of events. Then, test what you learned with the quiz questions.

Poisson Distributions

Imagine planning and taking a road trip with a few friends. The planned route has an average of two rest stops every 150 miles. Based on your car's mileage, you figure that the group need to stop for food and gas five times during the 600-mile trip. As one of your friends is a mathematician, you're curious to find the probability that the group will pass exactly five rest stops during the trip. To figure this out, you'll need to use a Poisson distribution.

Poisson distributions are used to calculate the probability of an event occurring over a certain interval. The interval can be one of time, area, volume or distance. You can find the probability of an event occurring is found using the formula in the Poisson distribution formula image.

Poisson distribution formula

Probability Conditions

There are two conditions that must be met in order to use a Poisson distribution. First, each successful event must be independent. In relation to our hypothetical road trip, the probability of finding one rest stop must not depend on finding another rest stop. Second, the probability of success over a short interval must equal the probability of success over a longer interval. For example, the probability of finding a rest stop is 2 per 150 miles and this rate must remain consistent for any length of distance.

Example One

Let's use the formula to find the probability that you will pass exactly five rest stops during your road trip. You can find the average number of stops per mile by dividing 2 by 150, which equals 0.013. For a 600-mile trip, you and your friends can expect 7.8 stops, the result of multiplying 0.013 by 600. The number of success we're looking for here is five. Substituting these values (7.8 and 5) into the formula, we get 0.099. This tells us that there is a 9.9% chance that you and your friends will pass exactly five rest stops during your 600-mile road trip.

Probability of passing five rest stops in 600 miles

Example Two

In this next example, we will see how the Poisson distribution can be used to find the amount of bacteria in a water sample.

  • A river has an average of 3 E. coli bacteria per 5 mL of water. What is the probability that exactly 20 bacteria are found in a 50 mL sample?

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