Count possible outcomes, update what you know, and separate an average from a promise.
6 original activities1 interactive model3 quick questions
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Four deliveries, many possible outcomes
Imagine four independent deliveries with the same on-time probability. Change that probability and see the exact distribution of on-time counts. This is a fictional model, not carrier data.
Every row is the probability of exactly that many on-time deliveries. The five probabilities total 100%, apart from display rounding.
Binomial model: P(k) = C(4,k) × pk × (1−p)4−k. Four deliveries; known probability p; independent outcomes.
Exact probabilities
0 of 4
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1 of 4
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2 of 4
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3 of 4
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4 of 4
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Predict. Try. Understand.
Pause at each question before opening the explanation. Examples are invented for learning unless stated otherwise.
IDEA 01
Count the paths, not just the labels
Are zero, one and two on-time deliveries equally likely for two fair yes/no outcomes?
TRY / THINK
List the possibilities for two deliveries: on/on, on/late, late/on and late/late. Group them by the number on time.
Explore the explanationWHY IT WORKS
If the two outcomes are independent and each is 50/50, the four paths are equally likely. The grouped totals are not: there are two different ways to get exactly one on-time delivery.
Worked example
P(0) = 1/4, P(1) = 2/4 and P(2) = 1/4. Equal-looking categories can contain different numbers of outcomes.
TAKE IT INTO PLANNING
A dashboard category is not automatically a probability. First understand what events are counted inside it.
IDEA 02
A smaller question changes the denominator
If you know a delayed parcel came by road, should you divide by all parcels or road parcels?
TRY / THINK
Imagine 20 parcels: 12 road and 8 rail. Four road parcels and one rail parcel are late. Find the late share overall, then within road.
Explore the explanationWHY IT WORKS
A conditional probability restricts the group you are considering. The numerator and denominator must describe that same restricted group. This describes the example; it does not establish that road transport caused the delays.
Worked example
Overall late share = 5/20 = 25%. Among road parcels it is 4/12, about 33.3%.
TAKE IT INTO PLANNING
Always name the denominator. “Percent late” becomes useful only when the population and period are clear.
IDEA 03
An average can be a number you never see
Can expected demand be 16 units if demand is always either 10 or 30?
TRY / THINK
Use a fictional day with 70% chance of 10 units and 30% chance of 30 units. Multiply each quantity by its probability and add.
Explore the explanationWHY IT WORKS
Expected value is a probability-weighted average. It describes a model over repeated cases, not a guarantee for the next case. A plan built only around the average may miss expensive high-demand outcomes.
Worked example
0.70 × 10 + 0.30 × 30 = 16 units. Neither possible day in this model has demand exactly 16.
TAKE IT INTO PLANNING
Compare the consequences of shortages and leftover stock alongside the expected value.
IDEA 04
Two suppliers can share one risk
Does using two suppliers always halve disruption risk?
TRY / THINK
Compare two fictional suppliers using different ports with two using the same port. Name a disruption that could affect both.
Explore the explanationWHY IT WORKS
Independence means learning one outcome gives no information about the other. Shared roads, power, upstream materials or weather can break that assumption. The distribution above uses independence deliberately; real operations may not satisfy it.
Worked example
If two independent suppliers each have a 10% failure probability, both fail with probability 0.10 × 0.10 = 1%. A shared shutdown can make that calculation inappropriate.
TAKE IT INTO PLANNING
Diversification requires different risk exposure, not just different supplier names.
IDEA 05
A short streak is weak evidence
Do three late deliveries prove a carrier always fails?
TRY / THINK
Compare a record of 3 late out of 3 with 300 late out of 1,000. Write down what each record does and does not tell you.
Explore the explanationWHY IT WORKS
The observed fraction summarizes the sample. It does not automatically reveal the future probability. Small samples fluctuate, and a selected sample of only complaints is not representative of all shipments.
Worked example
3/3 = 100% in the tiny sample; 300/1,000 = 30% in the larger sample. Neither figure is a universal guarantee about tomorrow.
TAKE IT INTO PLANNING
Review sample size, dates and selection before ranking suppliers.
IDEA 06
Rare per trial is not rare forever
If an event has a 1% chance each time, must it happen exactly once in 100 trials?
TRY / THINK
For 100 independent trials, calculate the chance that it never happens: 0.99 multiplied by itself 100 times.
Explore the explanationWHY IT WORKS
Counting the complement is often easier. One minus the probability of no occurrence gives the probability of at least one. Independence and an unchanged 1% probability are assumptions, not facts about every real process.
Worked example
1 − 0.99^100 is about 63.4%. It is neither 1% nor 100%, and does not say exactly one occurrence will happen.
TAKE IT INTO PLANNING
Repeated exposure can matter even when each individual risk looks small.
A SMALL CHECK, NO PRESSURE
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