Which station made the sampled fixture?
Find the chance a passing fixture came from Station A.
The question
Two stations in a fictional workshop make and check fixtures. The interviewer samples one passing fixture.
| Made | Passed | |
|---|---|---|
| Station A | 40 | 24 |
| Station B | 60 | 12 |
Each passing fixture has the same chance of selection.
“How would you explain the chance the fixture came from Station A?”
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A strong answer, step by step
Open each step in turn. This is one strong answer, not the only one.
Step 1: Which group is sampled?
What would you do here? Open the move you’d make.
Use all 100 fixtures made
Tempting, but it misses something. This includes 64 fixtures the sampler cannot select.
Use the 36 passing fixtures
The strong move. Only fixtures that passed can be selected.
Use the 40 fixtures made by A
Tempting, but it misses something. That measures how often A passes, given it made a fixture.
See how a strong candidate says it
“The sample comes from the 36 passing fixtures. I’ll count how many of those came from A.”
Why it works: All passes are equally likely. The total number made includes fixtures that cannot be selected.
Step 2: What is the chance of A?
What would you do here? Open the move you’d make.
3/5, from 24 of 40 fixtures
Tempting, but it misses something. That is the pass rate at A. The sample already contains only passes.
2/5, from 40 of 100 fixtures
Tempting, but it misses something. That would apply to a uniform sample from everything made.
2/3, from 24 of 36 passes
The strong move. The numerator is the passes from A. The denominator is all passes.
See how a strong candidate says it
“Station A supplied 24 of the 36 passes, so the chance is 24/36 = 2/3.”
Why it works: Each pass has the same weight under the opening rule.
Step 3: What is the chance now?
The sampler actually chooses one of the 2 stations uniformly, then one of that station’s passing fixtures uniformly. Both stations have passes.
What would you do here? Open the move you’d make.
1/2, from the station draw
The strong move. The sampler gives A and B equal weight at the first draw.
2/3, from 24 of 36 passes
Tempting, but it misses something. This would treat each pass as equally likely across both stations.
3/5, from A’s pass rate
Tempting, but it misses something. The second draw is already restricted to passes at the selected station.
See how a strong candidate says it
“It is 1/2. Once a station is selected, the sampler always picks a pass from that station.”
Why it works: The first draw determines the station. Each station has passes to choose from.
Step 4: Why did the answer change?
What would you do here? Open the move you’d make.
Say the stations now pass equally often
Tempting, but it misses something. Station A passed 24 of 40 and Station B passed 12 of 60. Those counts did not change.
Explain the changed sampling weights
The strong move. The selection rule accounts for the whole change.
Say every pass has the same weight
Tempting, but it misses something. A has more passes splitting its half of the chance.
See how a strong candidate says it
“The counts stayed the same. Each A pass now has chance 1/48. Each B pass has chance 1/24. Each station’s passes add up to 1/2 under this rule.”
Why it works: The new rule gives each station equal total weight, even though their passing counts differ.
The answer, in short
- Decision
- Use 2/3 for a uniform pass draw and 1/2 for a uniform station draw.
- Because
- The two rules give different weights to passing fixtures.
- What would change it
- Ask what the sampler does if a selected station has no passes.
Common traps
- Using the production total after conditioning on a pass
- Treating the pass rate at A as the chance of A
Check your answer
- What does the sampler select first?
- What is the denominator for the original draw?
- Which weights change in the new rule?
- What happens if a station has no passing fixture?
Last reviewed Nov 11, 2026