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DM topic guide

Probability

Calculate single, combined and conditional outcomes.

Guide page7 of 7

Set up a frequency table or probability tree before calculating — never run numbers blind.

The method — step by step

1
Name the type

Single event, combined AND/OR, without replacement, conditional (given that), or binomial (exactly k from n)?

2
Use a concrete base

Convert percentages to frequencies using 100, 1 000 or 10 000. Whole numbers are far easier to work with.

3
Build a table or tree

Tables work for two conditions. Trees work for sequential events where the denominator changes.

4
Track denominators

With replacement: denominator stays fixed. Without replacement: subtract one after every draw.

5
Restrict for conditional

'Given that it is defective' means work only within the defective sub-group — not the full sample.

Convert to a frequency table first

DefectiveOKTotal
Machine A1205,8806,000
Machine B2003,8004,000
Total3209,68010,000

P(B | defective) = 200 ÷ 320 = 62.5% — the answer comes straight off the table.

Use 100, 1,000 or 10,000 as your base — whichever turns the percentages into whole numbers with no messy rounding.

Worked example

EXAMPLEDefective components — which machine is to blame?

Given

A factory uses two machines to produce components. Quality control records show the following:
Machine A produces 60% of all components, of which 2% are defective.
Machine B produces the remaining 40%, of which 5% are defective.
An inspector picks a component at random from the day's output and finds it is defective.

Question

What is the probability that the defective component was made by Machine B?

Working

Step 1 — turn percentages into real counts (use 10,000 as the base)

MachineParts made% defectiveDefective count
Machine A6,0002%120
Machine B4,0005%200
Total10,000—320

Step 2 — we already know it's defective, so only look at those 320

Ignore all 9,680 non-defective parts — they're ruled out. Focus only on the 320 defective ones.

320 defective

Of those 320 defective parts: 120 came from Machine A, 200 came from Machine B.

120 + 200

Chance it came from Machine B → 200 out of 320 defective parts → 200 ÷ 320

= 62.5%
AnswerC — 62.5% probability it came from Machine B

Rules to remember

Set up before calculating

Write out the table or tree with labels before touching any numbers.

Without replacement: −1 each draw

Second draw from 12 items has denominator 11, not 12.

Conditional: restrict your sample

'Given that it is defective' means ignore non-defective components entirely.

At-least-one: use the complement

P(at least one) = 1 − P(none). It is almost always the faster route.

Binomial: C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ

Exactly k successes from n independent trials. C(n,k) counts the arrangements.

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