Unit 8 · Statistics & ProbabilityS5 · Conditional Prob. · Telemetry

Two-Way Tables & Conditional Probability

Choose the conditioned group first, then use the right denominator.

28 min
Unlocks: Fault Detection

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

conditional probability: Probability within a stated subgroup.
marginal total: A row or column total at the edge of a table.
joint count: A count meeting two conditions.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Underline the condition after “given” or “among.” Find that row or column total before looking at the numerator. Then take the overlap of the condition and target.

Key idea

The decision that matters

The word “given” shrinks the universe. Your denominator is the total of the given group.

Conditional probability

P(A\mid B)=\frac{n(A\cap B)}{n(B)}

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

Among 30 commuters who bike, 18 wear helmets. The probability that a randomly chosen biker wears a helmet is $\frac{18}{30}=\frac35$ .

Cue	Best move
Word after given	this is the denominator group
Two conditions	numerator is their overlap

Trap

Using the grand total as the denominator

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Condition on a row

Problem

A survey records 36 bus riders and 24 car riders. Of these, 15 bus riders and 6 car riders arrive before 8:00. Given that a selected person rides the bus, find the probability the person arrives before 8:00.

“Given bus” restricts the denominator to 36 bus riders.

The joint count for bus and before 8:00 is 15.

Which fraction conditions on bus riders?

The probability is $15/36=5/12$ .

Why is 36 the denominator in the first conditional probability?

Worked example

Example 2: Reverse the condition carefully

Problem

Using the same survey, find the probability a person rides the bus given that the person arrives before 8:00.

Now the conditioned group is everyone arriving before 8:00:

15+6=21

Of those 21, 15 ride the bus.

Which fraction conditions on early arrivals?

The probability is $15/21=5/7$ .

Why does reversing the condition change the denominator to 21?

Faded example · you finish it

Example 3: You finish it

Problem

Among 40 students taking art, 14 also take music. Find

P(\text{music}\mid\text{art})

Underline the group after “given”; that group supplies the denominator.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.