Shrinking Interval Slides Lesson 01
The Shrinking
Interval
Bridging the gap between discrete trials and continuous arrivals.
Poisson Processes Series
Spot the Difference
The Typo
A book has 400 pages. There are 200 typos distributed randomly. What's the chance of a typo on Page 42?
Discrete structure (Pages)
The Call
A call center gets 200 calls over 400 hours. What's the chance of a call in the next 10 minutes?
Continuous structure (Time)
Why Binomial Fails
Binomial models \( n \) independent trials with probability \( p \).
How many "trials" are in an hour of time?
Can two events happen at the exact same instant?
What happens if we split the hour into seconds? Milliseconds?
The Intuition
The Shrinking Limit
n = 4 intervals
n = 24 intervals
n → ∞
The Mathematical Leap
Let \( \lambda \) be the expected number of events. If we have \( n \) trials, then \( p = \frac{\lambda}{n} \). As \( n \to \infty \):
\[ P(X=k) = \lim_{n \to \infty} \binom{n}{k} \left(\frac{\lambda}{n}\right)^k \left(1 - \frac{\lambda}{n}\right)^{n-k} \]
\[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
When is it a "Poisson Process"?
Independence
The occurrence of one event doesn't affect the probability of another.
Constant Rate
The average rate \( \lambda \) stays the same over the entire interval.
Rarity
Events cannot happen at the exact same moment.
Is rain a Poisson process?
"Think about the independence of drops. If one drop hits the ground, does it make the next drop more or less likely? Does the rate change over 10 minutes?"
Shrinking Interval Discovery Worksheet Arrival Logic // 01
LAB: THE SHRINKING INTERVAL
NAME:
DATE:
Learning Objective
Differentiate between discrete trials and continuous time intervals, and derive the intuitive need for the Poisson distribution as the limit of the Binomial distribution.
I. The Grain of Reality
Consider two scenarios where events happen randomly. For each, identify the "trials" (independent opportunities for an event to happen).
Scenario A: Exam Errors
A teacher grades a 50-question multiple-choice test. There are 5 errors total.
What defines a "trial" here?
Scenario B: Meteor Strikes
A satellite monitors the atmosphere for 24 hours. There are 5 strikes total.
What defines a "trial" here?
Reflection Question:
In Scenario B, why is it problematic to say there are exactly 24 trials (one per hour)? What happens if two meteors strike in the same hour?
II. The Shrinking Interval Experiment
Imagine a busy bus stop. On average, \( \lambda = 4 \) buses arrive every hour. We want to find the probability that exactly 3 buses arrive in a specific hour.
Interval Size (\( n \)) Prob. of Event (\( p = \frac{4}{n} \)) Binomial: \( P(X=3) = \binom{n}{3} p^3 (1-p)^{n-3} \) \( n = 4 \) (15-min blocks) \( p = 1.00 \) \( \binom{4}{3} (1)^3 (0)^1 = 0.00 \) (Wait, why is this 0?) \( n = 10 \) (6-min blocks) \( p = 0.40 \) \( \binom{10}{3} (0.4)^3 (0.6)^7 \approx 0.2150 \) \( n = 60 \) (1-min blocks) \( p = 0.067 \) \( \binom{60}{3} (0.067)^3 (0.933)^{57} \approx 0.1988 \) \( n = 3600 \) (seconds) \( p = 0.0011 \) CALCULATING...
1. Why does the probability change as \( n \) increases?
2. What physical assumption are we making when we use larger \( n \)? (Hint: Think about two buses arriving at the same time.)
III. Towards Continuity
"As we divide time into infinitely many tiny slices..."
\[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
Check for Understanding:
If radioactive decay happens at a rate of 12 counts per minute, use the formula above to set up (don't solve) the calculation for the probability of observing 0 counts in one minute.
SHOW WORK HERE
© 2026 Arrival Logic Lab Series Lesson 01 / Page 1 of 2
Shrinking Interval Teacher Guide TEACHER GUIDE
LESSON 01: THE SHRINKING INTERVAL
Duration
50-60m
Pedagogical Approach
This lesson uses an inquiry-based limit approach . Instead of just handing students the Poisson formula, we force them to see why the Binomial distribution breaks down when applied to continuous time. By "shrinking the interval," students discover that the Poisson distribution isn't just a new formula—it's the logical conclusion of a Binomial process with an infinite number of trials.
Key Skills
Identifying discrete vs continuous
Conceptualizing limits (\( n \to \infty \))
Parameterization (\( p = \lambda/n \))
Facilitation Notes
The Hook: Typos vs. Calls
When discussing the hook, push students on overlap . In Scenario A, you can't have two "Question 1s" at the same time. But in Scenario B, can two calls happen at the exact same millisecond? Physically, maybe not, but mathematically, as we increase the precision of our measurement, the probability of "overlap" in a tiny slice of time goes to zero.
Worksheet Part II: The Paradox
Students will notice that for \( n=4 \), \( P(X=3) = 0 \). Ask: "Why does the math tell us it's impossible for 3 buses to arrive if we have 4 blocks of time?" Answer: The Binomial model assumes at most 1 event per trial. If we only have 4 trials and \( p=1 \), we can only ever have exactly 4 arrivals. The model is too "chunky" for reality.
The Derivation
You don't need to do the full calculus proof (unless they are a high-level BC Calculus group). Focus on the ingredients of the Poisson formula:
\( e \): Comes from the limit \( \lim (1 + x/n)^n \). It represents growth and continuous processes.\( \lambda \): The "anchor." It's the only thing we know (the average rate).\( k! \): Corrects for the fact that the order of arrivals doesn't matter.
Common Misconceptions
"Why not just use Binomial with large n?"
Students might think Poisson is just a "short cut." Emphasize that in continuous time, there is no "real" \( n \). Time has no smallest unit.
"Is lambda always an integer?"
Remind them \( \lambda \) is an average. You can have 4.2 buses per hour on average, even though you can't see 0.2 of a bus.
Arrival Logic Series // Lesson 01 Teacher Facilitation Guide
Rate of Things Slides Lesson 02
The Rate
of Things
Mastering the parameter \( \lambda \) and scaling probabilities across time.
Target: Skill-Building
The Poisson Engine
The Poisson distribution tells us the probability of seeing exactly \( k \) events in a given interval when the average rate is \( \lambda \).
\( \lambda \) Expected number of events per interval
\( k \) Actual number of events observed
\[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
Note: \( e \approx 2.71828 \)
The Elastic Interval
\( \lambda \) is tied to its time window. If the window changes, \( \lambda \) scales linearly.
Base Rate
\( \lambda = 6 \)
Events / Hour
Scaled Rate
\( \lambda = 1.5 \)
Events / 15 Minutes
"If you expect 6 calls in 60 minutes, you expect 1 call every 10 minutes."
Case Study: Radioactivity
Geiger Counter Data
A sample decays at an average rate of 30 counts per minute .
Question: What is the probability of seeing exactly 2 counts in the next 10 seconds?
1. Scale: \( \lambda = 30 / 6 = 5 \) counts per 10s
2. Identify: \( k = 2 \)
3. Calculate: \( P(X=2) = \frac{e^{-5} 5^2}{2!} \)
"At Least One"
Sometimes we don't want exactly \( k \). We want \( P(X \ge 1) \).
The Complement Rule
\[ P(X \ge 1) = 1 - P(X = 0) \]
\[ P(X \ge 1) = 1 - e^{-\lambda} \]
Quick Check
A bakery sells an average of 12 cookies every hour. What is the expected number of cookies sold in a 10-minute window?
A) 1.2
B) 2.0
C) 6.0
Rate of Things Problem Set Arrival Logic // 02
PROBLEM SET: SCALING THE RATE
NAME:
DATE:
Reference Box
\[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
1
The Call Center
A customer support line receives an average of 4.5 calls per hour . Calculate the probability that they receive exactly 2 calls in a single hour. (Round your final answer to 4 decimal places).
\( \lambda \) =
\( k \) =
Work Space:
2
The Particle Counter
A Geiger counter records an average of 120 radioactive decays per minute . We are interested in a 5-second window .
Step A: Scaling the Rate
What is the expected number of decays (\( \lambda \)) in 5 seconds?
Step B: Probability of Silence
What is the probability that zero decays occur during a 5-second interval? Show the setup and the result.
3
The Meteorologist's Paradox
In a specific region, major lightning strikes occur at a rate of 0.8 strikes per day .
A) What is the probability of at least one strike in a single day?
B) A safety inspector claims there is a 50% chance of seeing at least one strike in an 18-hour window. Test this claim. Is it accurate? (Show your math).
© 2026 Arrival Logic Lab Series Lesson 02 / Page 1 of 1
Rate of Things Answer Key ANSWER KEY
PROBLEM SET: SCALING THE RATE (L02)
1 The Call Center
\( \lambda \): 4.5
\( k \): 2
\( P(X=2) = \frac{e^{-4.5} \cdot 4.5^2}{2!} \)
Common error: Students may forget to divide by 2! or miscalculate \( e^{-4.5} \).
2 The Particle Counter
Step A: Scaling the Rate
\( \lambda_{5s} = 120 / (60 / 5) = 120 / 12 = 10 \text{ decays} \)
Step B: Probability of Silence
\( P(X=0) = \frac{e^{-10} \cdot 10^0}{0!} = e^{-10} \cdot 1 \)
3 The Meteorologist's Paradox
A) At least one strike per day
\( \lambda = 0.8 \)
B) Testing the 18-hour claim
\( \lambda_{18h} = 0.8 \cdot \frac{18}{24} = 0.8 \cdot 0.75 = 0.6 \)Verdict: The claim is inaccurate . The probability is actually approx 45.1%, not 50%.
Arrival Logic Series // Lesson 02 Answer Key // Confidential Teacher Resource
Waiting Game Slides Lesson 03
The Waiting
Game
From counting arrivals to measuring the silence in between.
Poisson → Exponential
Perspective Shift
Poisson View
"How many?"
Counting the number of events in a fixed window of time.
Discrete
Exponential View
"How long?"
Measuring the time between two consecutive arrivals.
Continuous
The Logical Bridge
If we wait longer than time \( t \) for the next event, it means exactly zero events happened in the interval \( [0, t] \).
\( P(T > t) = P(X = 0 \text{ in interval } t) \)
Wait, we know \( P(X=0) \)!
\( P(T > t) = e^{-\lambda t} \)
The Distribution
The Exponential distribution is used to model the "time to first event."
Probability Density Function (PDF)
\[ f(t; \lambda) = \lambda e^{-\lambda t} \]
Cumulative Distribution Function (CDF)
\[ P(T \le t) = 1 - e^{-\lambda t} \]
High prob
Long wait is rare
Waiting for a Text
You receive an average of 4 texts per hour . What is the probability you wait more than 30 minutes for the next one?
1. Rate (\( \lambda \)): 4 texts/hr
2. Time (\( t \)): 0.5 hr
3. Equation: \( e^{-(4)(0.5)} \)
4. Result: \( e^{-2} \approx 0.135 \)
"Only a 13.5% chance of waiting longer than half an hour."
Does the bus have a memory?
"If you've already been waiting 10 minutes for a bus that arrives every 15 minutes, does that mean it's 'due' to arrive in the next 5 minutes? Or is your expected wait time still 15 minutes?"
Preview of Lesson 04...
Waiting Game Exploration Guide Arrival Logic // 03
LAB: THE SILENCE BETWEEN EVENTS
NAME:
I. The Poisson-Exponential Link
Recall the Poisson distribution: \( P(X=k) = \frac{e^{-\lambda t} (\lambda t)^k}{k!} \)
The "waiting time" \( T \) until the next event is greater than \( t \) if and only if there are exactly zero events in the interval \( [0, t] \).
1. Use the Poisson formula to write the probability that \( X = 0 \) in an interval of length \( t \):
P(X=0) = ...
2. This result is the probability that the waiting time \( T \) is greater than \( t \). We call this the "Survival Function":
\( P(T > t) = e^{-\lambda t} \)
3. Therefore, the probability that the event happens within time \( t \) (the CDF) is:
P(T ≤ t) = 1 - ...
II. Workshop: The Coffee Shop
Customers arrive at "The Infinite Grind" at an average rate of 10 customers per hour (\( \lambda = 10 \)).
Scenario A: The Short Wait
What is the probability that the next customer arrives within the next 5 minutes ?
Calculation steps:
Convert \( t \) to hours: \( 5 / 60 = 0.0833 \)
Apply CDF: \( 1 - e^{-10(0.0833)} \)
Scenario B: The Long Silence
What is the probability that the staff has to wait more than 15 minutes for a customer?
Calculation steps:
Convert \( t \) to hours
Apply Survival Function: \( e^{-\lambda t} \)
Challenge: The Median Wait
The median wait time is the value of \( t \) where \( P(T \le t) = 0.5 \).
III. Synthesis
If the rate \( \lambda \) increases (e.g., from 10 to 50 arrivals per hour), what happens to the shape of the Exponential curve?
Draw or describe the change in the PDF curve...
© 2026 Arrival Logic Lab Series Lesson 03 / Page 1 of 1
Waiting Game Answer Key ANSWER KEY
LAB: THE SILENCE BETWEEN EVENTS (L03)
I. The Poisson-Exponential Link
1. \( P(X=0) = \frac{e^{-\lambda t} (\lambda t)^0}{0!} = e^{-\lambda t} \)
II. Workshop: The Coffee Shop
Scenario A: The Short Wait
\( t = 5/60 = 0.0833 \)
Scenario B: The Long Silence
\( t = 15/60 = 0.25 \)
Challenge: The Median Wait
\( 1 - e^{-10t} = 0.5 \)
III. Synthesis
If \( \lambda \) increases (e.g., from 10 to 50), the curve decays much more rapidly . This means that the probability of very short waits increases, and the probability of longer waits becomes almost zero very quickly. The graph is "compressed" against the y-axis.
Arrival Logic Series // Lesson 03 Answer Key // Confidential Teacher Resource
Memoryless Property Slides Lesson 04
Clean Slate
Waiting
Investigating the paradoxical "Memoryless Property" of continuous arrivals.
Resetting the clock...
The Gambler's Fallacy
"I've been waiting for the bus for 20 minutes already. It's supposed to come every 15 minutes... it MUST be arriving any second now!"
Human Intuition: It's "Due"
"The fact that it hasn't arrived in 20 minutes tells me nothing about when it will arrive in the future."
The Math: It's Memoryless
Formal Definition
A distribution is memoryless if the probability of an event happening in the next \( t \) seconds is the same, regardless of how long you've already waited.
\( P(T > s + t \mid T > s) = P(T > t) \)
"Wait \( t \) more..." "Given already waited \( s \)" "Same as starting over"
Why it works
Remember our "Survival Function" for the Exponential Distribution:
\( P(T > t) = e^{-\lambda t} \)
The math follows the rule of exponents:
\( e^{-\lambda(s+t)} = e^{-\lambda s} \cdot e^{-\lambda t} \)
Calculus Note
The Exponential distribution is the only continuous distribution that has this property. It is perfectly consistent and perfectly stubborn.
The Inspection Paradox
If buses arrive according to a Poisson process with \( \lambda = 15 \) minutes, the average wait time for a passenger arriving at a random time is... 15 minutes .
Wait, shouldn't it be half the average (7.5m)?
No! Because the process doesn't "know" how long it's been since the last bus.
YOU ARRIVE
"The interval you land in is, on average, longer than the typical interval."
Is it Due?
We're going to look at several real-world scenarios. Your task: Decide if they are Poisson (Memoryless) or Non-Poisson (Memory) .
Start Investigation
Is It Due Discussion Cards CASE 01
The City Bus
Buses are scheduled to arrive every 15 minutes. One passenger has been waiting for 20 minutes.
Discussion Prompt
Is the next arrival independent of the previous one? Does the bus have a "memory" of being late?
CASE 02
Radioactive Decay
An atom of Uranium-238 has not decayed for 4 billion years.
Discussion Prompt
Is it "due" to decay? Or is the probability of decaying in the next second exactly the same as it was 4 billion years ago?
CASE 03
The Slot Machine
A machine hasn't paid out a jackpot in over 48 hours of continuous play.
Discussion Prompt
People often flock to "cold" machines thinking they are due. Is this a Poisson process? Why or why not?
CASE 04
Emergency Calls
A dispatcher hasn't received a call in 45 minutes during a quiet night shift.
Discussion Prompt
Does the length of the silence change the probability of a call arriving in the next 5 minutes?
Cut along the grid lines. Distribute to groups for a 10-minute deliberation.
Virtual Queues Slides Lesson 05
Virtual Queues
Building and testing Poisson processes through digital simulation.
root@stats-lab:~# ./simulate_arrivals
Beyond the Formula
The formulas give us averages and probabilities. Simulation gives us realizations .
"How many workers do I need right now?"
"What is the longest line that might form?"
"How often does the system overflow?"
// Poisson Generation Logic
1. Start at time \( T = 0 \)
Inverse Transform
To turn a computer's uniform random number (\( 0 \) to \( 1 \)) into an Exponential wait time, we use the Inverse CDF.
\[ Wait = -\frac{1}{\lambda} \ln(Random) \]
"The logarithm stretches the uniform noise into a curve where small values are common and large values are rare—perfectly matching the Exponential PDF."
Project Mission
Model
Generate 100 arrivals for a coffee shop with \( \lambda = 12 \) per hour.
Analyze
Calculate the average wait time and the maximum gap between customers.
Validate
Compare your simulation's mean to the theoretical \( 1/\lambda \).
Simulation Debugging
Unit Mismatch
If \( \lambda \) is arrivals per hour , your simulated time will also be in decimal hours (e.g., 0.5 = 30 mins).
Absolute Time
Don't confuse "Time Since Last" (the wait) with "Time Of Arrival" (the cumulative sum of waits).
Deploy!
"Your simulation data will be the foundation of your final lab report. Can your coffee shop survive a peak hour rush?"
Get Project Guide
Virtual Queues Project Guide Arrival Logic // 05
PROJECT: THE INFINITE GRIND SIMULATION
Project Goal
Construct a stochastic simulation of a Poisson arrival process. You will generate arrival data, analyze queueing metrics, and validate your simulated data against theoretical Poisson and Exponential expectations.
I. Data Generation
Use a spreadsheet (Excel/Sheets) or a simple script to generate 100 arrivals. Assume an average rate of \( \lambda = 15 \) customers per hour .
# Formula for cell A2 (Wait Time in Hours):
= - (1 / 15) * LN(RAND())
# Formula for cell B2 (Arrival Time):
= B1 + A2
Column A: Wait Times
This column represents the gap between customers. It should follow an Exponential distribution.
Column B: Arrival Times
This column is the running sum of wait times. It represents the absolute time each customer walks in.
II. Technical Analysis
Fill in the following metrics based on your 100-arrival simulation.
1. Sample Mean Wait Time (hours):
2. Theoretical Mean (1/\( \lambda \)):
3. The Rush Hour Check:
Count how many customers arrived between Hour 1 and Hour 2 of your simulation. (This is a Poisson count).
Number of Arrivals:
\( \lambda = 15 \text{ expected} \)
4. Maximum Wait:
What was the single longest gap between customers? How does this compare to the average?
III. Business Recommendation
Imagine you are the manager. If it takes 4 minutes to make a coffee, does your simulation suggest any "danger zones" where the queue would grow faster than the staff could handle?
Analyze the frequency of wait times that were less than 4 minutes...
Submission Checklist
Spreadsheet file (.csv / .xlsx)
Histogram of Wait Times
Comparison Table (Sim vs Theory)
Written Recommendation (Part III)
© 2026 Arrival Logic Lab Series Lesson 05 / Simulation Project
