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Mathematics
Conditional probability and Bayes
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Explore probability with a quality scanner: select groups, change the defect rate, and watch the denominator.
B · Flagged · Given condition: 1,350
P(A | B) = 900 / (900 + 450) = 66.7%
Bars: % of 10,000 products · Bulbs are illustrative
Sample: no flagged products yet
Select a tray to change the denominator
Set up the batch
Observed sample
Follow each product through the scanner. Parameter changes start a new sample.
Sampled0
Left in this run: 0
Representation
Change the denominator
P(A | B)
66.7%
- Both events
- 900
- Given condition
- 1,350
2 × 2 table
| Total | |||
|---|---|---|---|
| 1,000 | |||
| 9,000 | |||
| Total | 1,350 | 8,650 | 10,000 |
Select a group to change the denominator
Same numerator, different denominators
P(A | B) = 900 / 1,350 = 66.7%P(B | A) = 900 / 1,000 = 90%
P(A | B) considers flagged products; P(B | A) considers defective products.
Bayes: multiply along branches, add paths to B. P(A | B) = P(A)·P(B | A) / [P(A)·P(B | A) + P(Ā)·P(B | Ā)]
Model notes
- A means genuinely defective; B means flagged by the scanner. These hypothetical parameters do not describe an actual factory.
- The 10,000 products are normalized expected counts. Four equally sized cards show exact counts and linear bars sharing a 10,000-product scale. Large bulbs illustrate status, not quantity.
- Samples are independent draws from the same distribution, equivalent to sampling with replacement. Sample frequencies may differ from probabilities; error need not decrease at every step.
- A zero-probability condition gives an undefined conditional probability. An empty conditioned sample also has no corresponding frequency.
- Changing parameters clears the sample. Reset repeats the same pseudorandom sequence; speed changes time only: 1× takes 0.7 seconds per product, with 4× selected by default. Sample counts are labelled separately from expected population counts.