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Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Convert an arrival rate into an interval mean.
- Evaluate Poisson probabilities for specified counts.
- Recognize assumptions that separate the model from arbitrary count data.
Before you start
Exponentials, factorials, probability distributions, and time-unit conversion.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A fictional help desk receives requests according to a homogeneous Poisson process averaging two requests per hour. What are the probabilities of zero requests and exactly three requests in one hour?
Why this math matters
A count model links an average event rate to probabilities for an interval. Its parameter must match that interval: two requests per hour does not mean two per half hour. A Poisson process can model random arrivals under particular conditions, but bursty demand, scheduled arrivals, or changing hourly rates may need another model.
Set up the model
A useful answer starts with clear assumptions:
- The arrival rate is constant at two per hour over the period studied.
- Counts in disjoint intervals are independent under the process model.
- Arrivals occur singly in the Poisson model; a very short interval has negligible probability of multiple arrivals relative to its length.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
How does changing the time window change a count probability?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional help desk receives requests according to a homogeneous Poisson process averaging two requests per hour. What are the probabilities of zero requests and exactly three requests in one hour?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Set the interval parameter
λ = rate × time = 2 per hour × 1 hour = 2
λ is an expected count, a dimensionless number. It is not the probability of an arrival and can be larger than one.
Calculate no arrivals
P(X = 0) = e⁻²2⁰/0! = e⁻² ≈ 0.1353
Both 2⁰ and 0! equal one. Zero arrivals remain possible even when the average is two.
Calculate exactly three arrivals
P(X = 3) = e⁻²2³/3! = e⁻²(8/6) ≈ 0.1804
The formula gives one count's probability. At least three would require including all higher counts or using the complement of zero, one, and two.
The result
The one-hour probabilities are approximately 13.53% for zero requests and 18.04% for exactly three.
For a half-hour interval, λ becomes one. A homogeneous Poisson process has mean and variance equal to λ for its interval count. Noticeably different observed variation can be a reason to question the model rather than force the data into it.
Common mistakes to catch
- Leaving λ = 2 when switching to a half-hour interval uses the wrong expected count.
- An average arrival rate alone does not establish independent increments or a constant-rate Poisson process.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the probability of zero arrivals in half an hour?
Show a hint
Use λ = 1 for the shorter interval.
Reveal answer and explanation
e⁻¹ ≈ 36.79%
The expected half-hour count is one, and the zero-count probability is e⁻¹.
Practice 2
What is the probability of at least one arrival in one hour?
Show a hint
Take the complement of the zero-arrival case.
Reveal answer and explanation
1 − e⁻² ≈ 86.47%
Every nonzero count qualifies, so subtract 0.1353 from one.
Take the idea with you
Write rate units next to the time window before selecting a distribution parameter.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: How can today's state shape tomorrow's probabilities?
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