Where this fits
in engineering
Every bracket is inspected twice before it is packed. Neither check catches everything, and a firm that cannot predict its scrap cannot quote a job properly.
Two
Checks
Ashfield ships 1,000 steel brackets a week. Every one is inspected twice, and the two checks are independent. Passing the first tells you nothing about the second.
Start with the first check
Add the second check to every outcome
Multiply along each path
P(a bracket is rejected)
+ 0.019
+ 0.001
That is a scrap bill, not a rounding error.
Three things to carry into the exam
- Branches from one point add to 1
- Multiply along a path, add down the paths
- “At least one” is usually 1 − (none of them)
1 − 0.9506 = 0.0494
≈ 49 brackets
The engineering
context and termsAshfield fabricates steel brackets in batches of a thousand. Nothing comes off a shop floor perfect, so every bracket is inspected twice before it is packed: once for size, once for the weld. Neither check catches everything. A firm that cannot say how many parts it expects to scrap cannot quote a job properly, and probability is how an engineer turns two inspection pass rates into a number the production office can actually plan around.
- Bracket
- A small fabricated steel part, usually an L shape, used to join or support other components.
- Dimensional check
- An inspection that measures the part against the drawing to confirm the sizes are within tolerance.
- Weld integrity check
- An inspection of the welded joint, looking for cracks, porosity or poor fusion that would weaken it.
- Pass rate
- The proportion of parts that get through a given check, written here as a probability between 0 and 1.
- Scrap
- Parts rejected at inspection. They are reworked or thrown away, and both cost money.
Questions
4 questions · 12 marks-
A press passes 0.96 of the parts it makes. Two parts are pressed, independently. Find the probability that both pass. [2]0.96 × 0.96 = 0.9216
1 mark for multiplying, 1 for the answer. -
Using the figures in the video, find the probability that a bracket fails exactly one of the two checks. [2]0.049 + 0.019 = 0.068
Both failing routes needed. A common error is giving 0.069, which also counts the bracket that fails both. -
Two brackets are taken at random. Each is rejected with probability 0.069. Find the probability that at least one of the two is rejected, to 3 decimal places. [3]1 − (0.931 × 0.931)
= 1 − 0.866761
= 0.133
Recognising “at least one” as 1 − P(neither) earns the method mark even if the arithmetic slips. -
A batch of 2,500 brackets is made. The dimensional check passes 0.98 and a new weld process passes 0.99. Estimate how many brackets in the batch are rejected, and state one assumption you have made. [5]P(passes both) = 0.98 × 0.99 = 0.9702
P(rejected) = 1 − 0.9702 = 0.0298
0.0298 × 2500 = 74.5, so about 75 brackets
Assumption: the two checks are independent, and the pass rates apply equally to every bracket in the batch. Accept either. Answer must be a whole number of brackets with the unit stated.
Controls. Tap the stage or press space to play and pause. Drag the amber circle along the line to scrub, or use the arrow keys to jump five seconds. The chapter buttons drop you straight into each stage of the working, which is the bit worth pausing on with a class. Captions start off; CC turns them on. Everything is inside this one file, so it runs offline. Screen-record the stage at 1920×1080 if you need an MP4 with a voiceover.
Demo developed by Lee Guthrie for EAL