Bayesian updating in business
Use a frequency table to revise a demand or delivery forecast, then test whether the new probability changes the decision.
Update a prior probability with new evidence using a simple frequency table and explain what the revised belief means for a decision.
Should the owner book a second room?
A ceramics studio has until noon to reserve a second room for Saturday's workshop. Usually one room is enough. This morning the owner sees unusually heavy traffic on the booking page. Some of those visitors will buy seats; some are just looking. How much should she change her demand estimate, and is that change enough to book the room?
Both businesses and all figures in this lesson are fictional. A calculator and paper are enough.
Recall Lesson 038: Among 50 tile buyers, 40 had requested a sample. Among 135 sample requesters, those same 40 bought. Which fraction would you use to forecast a purchase after seeing a sample request?
Start with what you knew yesterday
The studio's prior probability is its estimate before seeing today's traffic. The owner uses 20% for the chance that demand will exceed one room's capacity. She based it on comparable Saturday workshops with the same price, season, and booking window. The other 80% represents ordinary demand. These two possibilities cover every workshop in the model.
The new evidence is a specific signal: booking-page traffic exceeded a preset count by 10 a.m. To interpret it, the owner needs two likelihoods: how often that signal appears on high-demand dates and how often it appears on ordinary dates. Assume those rates are 75% and 12.5%, respectively. If the signal were equally common in both groups, it would give no reason to change the 20% estimate.
The posterior probability is the revised estimate after observing the signal. Bayesian updating combines the prior with the likelihoods. Penn State's probability course describes Bayes' theorem as a way to calculate the probability of an event given evidence when the reverse conditional probabilities are known. A frequency table does the arithmetic without requiring you to memorize the formula. Penn State
Build the table from the assumptions
Imagine 1,000 comparable workshops. This is a convenient scale for counting, not a claim that the studio ran 1,000 workshops. The prior assigns 200 to high demand and 800 to ordinary demand. Apply each traffic rate to its own group: 75% of 200 = 150; 12.5% of 800 = 100. The remaining workshops have no signal.
| Demand | Heavy traffic | No heavy traffic | Total |
|---|---|---|---|
| High | 150 | 50 | 200 |
| Ordinary | 100 | 700 | 800 |
| Total | 250 | 750 | 1,000 |
Today the owner sees heavy traffic, so read only that column. Of the 250 workshops it represents, 150 have high demand. The posterior is 150 ÷ 250 = 60%. Today's evidence moved the estimate from 20% to 60%. The 75% used to build the table answered a different question: how often heavy traffic appears when demand is high.
If traffic had stayed below the cutoff, the owner would instead use 50 ÷ 750, about 6.7%. As a check, the two columns put 150 + 50 = 200 high-demand dates back into the original 1,000. An update changes which group best represents today. It does not turn visitors into paying customers.
Find the probability that changes the booking
The extra room costs $240 regardless of attendance. If demand is high, it lets the studio earn an additional $600 after all other additional costs but before the room fee. If demand is ordinary, the extra room earns nothing. Assume the studio can fund the fee and that booking the room does not change demand.
At the prior, the expected gain from booking instead of declining is 0.20 × $600 − $240 = −$120. At the posterior, it is 0.60 × $600 − $240 = $120. The break-even probability is $240 ÷ $600 = 40%. The evidence moved the estimate across that threshold, so book the room under these assumptions. There is still a 40% chance of ordinary demand and losing the $240 fee.
Check where the evidence came from. If a dashboard and a colleague's email both report the same traffic spike, count it once. A genuinely new signal can support another update, but you need to know how informative it is among workshops that already had heavy traffic. Reusing the first signal's rates can make the owner more confident without giving her any new information.
Exercise: Reserve backup freight?
Allow 8–10 minutes. You manage purchasing for an appliance manufacturer. Before a weekly dispatch report arrives, you estimate a 10% chance that a particular shipment will miss Friday's deadline. The report flags 80% of shipments that ultimately miss the deadline and 20% of those that arrive on time. It flags today's shipment.
For $250 you can reserve backup freight now. If the original shipment is late, the backup avoids $1,000 of additional cost. If it is on time, the backup saves nothing. The $250 is paid either way if you reserve. Assume sufficient cash, no other consequences, and that the backup does not affect the original shipment's timing.
- Build a frequency table for 1,000 hypothetical shipments. Compute the revised probability of lateness after the flag.
- Calculate the break-even late probability for the backup. Compare the decision before and after the flag.
- A colleague says that for this route the prior should be 5% instead of 10%. Hold the two flag rates fixed, rebuild the flagged column, and decide again.
- State which records you would check before accepting the colleague's change. Would a second copy of the same dispatch report settle the issue?
Try the calculations before reading the answer.
Answer
| Original shipment | Flag | No flag | Total |
|---|---|---|---|
| Late | 80 | 20 | 100 |
| On time | 180 | 720 | 900 |
| Total | 260 | 740 | 1,000 |
The 10% prior gives 100 late shipments. Of those, 0.80 × 100 = 80 are flagged. The 900 on-time shipments produce 0.20 × 900 = 180 flags. Given today's flag, the chance of lateness is 80 ÷ 260 ≈ 30.8%. Most flagged shipments still arrive on time.
The backup breaks even at $250 ÷ $1,000 = 25% late probability. Before the report, its expected net saving is 0.10 × $1,000 − $250 = −$150. After the flag, it is (80 ÷ 260) × $1,000 − $250 ≈ $57.69. Reserve the backup. Lateness does not have to be the most likely outcome for protection to be worthwhile.
At a 5% prior, only 50 of 1,000 shipments are late. The flagged column now contains 0.80 × 50 = 40 late and 0.20 × 950 = 190 on-time shipments. The posterior is 40 ÷ 230 ≈ 17.4%. Expected net saving is (40 ÷ 230) × $1,000 − $250 ≈ −$76.09. Decline the backup under the revised assumptions. The same flag raises both priors, but only one revised estimate clears 25%.
Check dispatch and delivery records for comparable shipments on this route, including unflagged shipments. Count misses against the same Friday deadline. Check the sample size and whether the carrier or reporting rule changed. The duplicate report adds no evidence about the route's base rate.
Recall answer: Use 40 ÷ 135, about 29.6%, because the request is what you observed. Today's tables take that same denominator rule one step further: you build the counts from a starting probability and the signal rates rather than receiving a completed table.
Applying this lesson
For your decision tree, record four numbers together: the prior, the signal rate in each of the two outcome groups, and the posterior after the observed signal. Put the action threshold beside them. That shows a reviewer both why the estimate changed and whether the change was large enough to matter.
Sources
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