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Probability: mixed assessment

Read the idea, work independently, then explain what changed.

TOPIC 01

Probability: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

A fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.

k=7

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Describe the sample space and event sets, then count or apply a probability identity.

Hint 2

Count favourable elementary outcomes.

Worked solution

Describe the sample space and event sets, then count or apply a probability identity.
Calculate or simplify this relation.
$P=1/10$
Equal-likelihood is stated, not inferred merely from the labels.

The requested value is 0.1.

Checks and common pitfalls: Equal-likelihood is stated, not inferred merely from the labels.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Two independent components each work with probability k/(k+1). A series system needs both. Find success probability.

k=7

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use independence only when justified and keep it distinct from disjointness.

Hint 2

Series success is an intersection.

Worked solution

Use independence only when justified and keep it distinct from disjointness.
Calculate or simplify this relation.
$P=(k/(k+1))^2=49/64$
“At least one” would describe a different system.

The requested value is 0.765625.

Checks and common pitfalls: “At least one” would describe a different system.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

A model has p=1/4. In 20k independent trials, find the expected number of occurrences.

k=7

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Expected count equals trial number times model probability.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$E(N)=np=140/4=35$
The actual integer count need not equal its expectation.

The requested value is 35.

Checks and common pitfalls: The actual integer count need not equal its expectation.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Disjoint events have probabilities 1/(k+3) and 2/(k+3). Find their union probability.

k=8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Describe the sample space and event sets, then count or apply a probability identity.

Hint 2

Disjointness removes the overlap term.

Worked solution

Describe the sample space and event sets, then count or apply a probability identity.
Calculate or simplify this relation.
$P(A\cup B)=1/11+2/11=3/11$
Mutually exclusive does not mean independent.

The requested value is 0.27272727.

Checks and common pitfalls: Mutually exclusive does not mean independent.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

A and B are disjoint with P(A)=P(B)=1/(k+2). Are they independent?

k=8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use independence only when justified and keep it distinct from disjointness.

Hint 2

Compare the intersection probability with the product of probabilities.

Worked solution

Use independence only when justified and keep it distinct from disjointness.
Calculate or simplify this relation.
$0=P(A\cap B)\ne P(A)P(B)=1/(k+2)^2$
Positive-probability disjoint events are dependent.

No, their intersection probability is zero but the product of their probabilities is positive.

Checks and common pitfalls: Positive-probability disjoint events are dependent.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A program’s pseudo-random outputs are reused with the same seed. Does that create new independent evidence each rerun?

n=80

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Inspect whether the sequence actually changes.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$\text{same seed}\Rightarrow\text{same deterministic sequence}$
Reproducibility is valuable but does not multiply the number of independent runs.

No; reproducing the same sequence repeats the same simulated observations.

Checks and common pitfalls: Reproducibility is valuable but does not multiply the number of independent runs.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

A claim assigns probability 1+1/k to an event. Explain why it cannot be valid.

k=9

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Describe the sample space and event sets, then count or apply a probability identity.

Hint 2

Check the probability axioms before calculating.

Worked solution

Describe the sample space and event sets, then count or apply a probability identity.
Calculate or simplify this relation.
$1+1/k>1$
All event probabilities must lie in [0,1].

It exceeds one, outside the allowed probability range.

Checks and common pitfalls: All event probabilities must lie in [0,1].

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Independent A,B have probabilities 1/2 and 1/(k+2). Find P(A∩B).

k=9

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use independence only when justified and keep it distinct from disjointness.

Hint 2

Independent event probabilities multiply.

Worked solution

Use independence only when justified and keep it distinct from disjointness.
Calculate or simplify this relation.
$P(A\cap B)=1/2\cdot1/11=1/22$
The multiplication is supported by the independence assumption.

The requested value is 0.04545455.

Checks and common pitfalls: The multiplication is supported by the independence assumption.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Two batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.

k=9

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Pool counts and trial totals before dividing.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$f=(18+27)/(45+90)=1/3$
A simple average of batch percentages ignores their unequal sizes.

The requested value is 0.33333333.

Checks and common pitfalls: A simple average of batch percentages ignores their unequal sizes.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

P(A)=k/(k+2). Find P(Aᶜ).

k=10

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Describe the sample space and event sets, then count or apply a probability identity.

Hint 2

An event and its complement partition the sample space.

Worked solution

Describe the sample space and event sets, then count or apply a probability identity.
Calculate or simplify this relation.
$P(A^c)=1-10/12=2/12$
The complement is relative to the stated sample space.

The requested value is 0.16666667.

Checks and common pitfalls: The complement is relative to the stated sample space.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

A,B are independent and P(B)>0. Compare P(A|B) with P(A).

P(B)=1/11

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use independence only when justified and keep it distinct from disjointness.

Hint 2

Use the product definition of independence.

Worked solution

Use independence only when justified and keep it distinct from disjointness.
Calculate or simplify this relation.
$P(A|B)=P(A\cap B)/P(B)=P(A)$
Conditioning on B does not change A’s probability in this model.

They are equal.

Checks and common pitfalls: Conditioning on B does not change A’s probability in this model.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

A simulation makes 10k equally likely selections from labels 1,…,k. What model probability does label 1 have?

k=10

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Separate the per-trial model from the number of simulations.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$p=1/10$
More simulations improve empirical evidence; they do not alter the specified probability.

The requested value is 0.1.

Checks and common pitfalls: More simulations improve empirical evidence; they do not alter the specified probability.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: A fair coin is tossed twice. List the event “exactly one head”. Part B: A model has p=1/4. In 20k independent trials, find the expected number of occurrences.

\begin{gathered}\text{B: }k=7\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Describe the sample space and event sets, then count or apply a probability identity.

Hint 2

B: Distinguish observed relative frequency from a model probability and describe the experiment.

Worked solution

Part A reasoning
Describe the sample space and event sets, then count or apply a probability identity.
Calculate or simplify this relation.
$A=\{HT,TH\}$
An event is a set; its probability is a number.
Part B reasoning
Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$E(N)=np=140/4=35$
The actual integer count need not equal its expectation.

A: {HT,TH}. B: The requested value is 35.

Checks and common pitfalls: An event is a set; its probability is a number. The actual integer count need not equal its expectation.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

The probability of each component failing is 1/(k+1), but common power failures can affect both. Can the independent-series formula be used without further evidence?

k=11

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use independence only when justified and keep it distinct from disjointness.

Hint 2

Inspect shared causes, not just marginal probabilities.

Worked solution

Use independence only when justified and keep it distinct from disjointness.
Calculate or simplify this relation.
$P(A\cap B)\ne P(A)P(B)\text{ in general}$
A numerical marginal rate does not establish independence.

No; a shared cause may create dependence.

Checks and common pitfalls: A numerical marginal rate does not establish independence.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

An event occurs 3k times in 10k trials. Find its relative frequency.

k=11

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish observed relative frequency from a model probability and describe the experiment.

Hint 2

Use occurrences divided by trials.

Worked solution

Distinguish observed relative frequency from a model probability and describe the experiment.
Calculate or simplify this relation.
$f=33/110=0.3$
Observed frequency is data, not an exact guarantee of future outcomes.

The requested value is 0.3.

Checks and common pitfalls: Observed frequency is data, not an exact guarantee of future outcomes.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

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Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.