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Probability and statistics · eight-question check

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

TOPIC 01

Probability and statistics

Try two tasks per skill independently, then revisit the linked skill.

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

Of 50 students, 20 cycle and 8 of those wear glasses. Given cycling, what is P(glasses)?

N=50; n(C)=20; n(G∩C)=8N=50;\ n(C)=20;\ n(G\cap C)=8
  • Select uniformly among the named people; the conditioning group is nonempty.
Skills and prerequisite lessons

Scope: JM01 / JM02 · This task uses elementary probability rules or polynomial calculus named in the JAE syllabus.

  1. 0.16
  2. 0.4
  3. 2.5
  4. 0.8
Working and explanation
BUILD THE REASONING
Hint 1
Restrict to the cycling group.
Hint 2
Start from this intermediate relation.
P(G∣C)=8/20P(G|C)=8/20
Worked solution
  1. Restrict to the cycling group.
  2. Establish and simplify this mathematical relation.
    P(G∣C)=8/20P(G|C)=8/20
  3. Establish and simplify this mathematical relation.
    P(G∣C)=0.4P(G|C)=0.4
  4. The denominator represents given information.
0.4
Checks and common pitfalls: The denominator represents given information.
Think first. Reveal a hint when the class is ready.
02 / Foundation#Your turn
A,B are disjoint and each has probability 1/4. Are they independent?
P(A)=P(B)=1/4; A∩B=∅P(A)=P(B)=1/4;\ A\cap B=\varnothing
  • Use P(A∩B)=P(A)P(B) as the independence criterion, including the stated overlap or disjointness.
Skills and prerequisite lessons
Scope: JM01 / JM02 · This task uses elementary probability rules or polynomial calculus named in the JAE syllabus.
  1. Yes: disjoint means independent.
  2. Yes: equal probabilities imply independence.
  3. No: 0 differs from 1/16.
  4. Impossible to decide.
Working and explanation
BUILD THE REASONING
Hint 1
Compare intersection and product.
Hint 2
Start from this intermediate relation.
P(A∩B)=0P(A\cap B)=0
Worked solution
  1. Compare intersection and product.
  2. Establish and simplify this mathematical relation.
    P(A∩B)=0P(A\cap B)=0
  3. Establish and simplify this mathematical relation.
    P(A)P(B)=1/16P(A)P(B)=1/16
  4. Positive-probability disjoint events are dependent.
No: 0 differs from 1/16.
Checks and common pitfalls: Positive-probability disjoint events are dependent.
Think first. Reveal a hint when the class is ready.
03 / Foundation#Your turn
A fair X takes 0,2. What is Var(X)?
X=(0,2); P=(1/2,1/2)X=(0,2);\ P=(1/2,1/2)
  • Finite-distribution probabilities must be nonnegative and sum to one; dependence is only as explicitly stated.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. 2
  2. 0
  3. 1
  4. 4
Working and explanation
BUILD THE REASONING
Hint 1
Center before squaring or subtract the squared mean from the second moment.
Hint 2
Start from this intermediate relation.
E(X)=1; E(X2)=2E(X)=1;\ E(X^2)=2
Worked solution
  1. Center before squaring or subtract the squared mean from the second moment.
  2. Establish and simplify this mathematical relation.
    E(X)=1; E(X2)=2E(X)=1;\ E(X^2)=2
  3. Establish and simplify this mathematical relation.
    Var⁡(X)=2−1=1\operatorname{Var}(X)=2-1=1
  4. The variance is not the raw second moment.
1
Checks and common pitfalls: The variance is not the raw second moment.
Think first. Reveal a hint when the class is ready.
04 / Foundation#Your turn
Var(X)=2. What is Var(3X+5)?
Var⁡(X)=2\operatorname{Var}(X)=2
  • Finite-distribution probabilities must be nonnegative and sum to one; dependence is only as explicitly stated.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. 11
  2. 18
  3. 6
  4. 23
Working and explanation
BUILD THE REASONING
Hint 1
Square the scale and remove the constant shift.
Hint 2
Start from this intermediate relation.
Var⁡(3X+5)=9Var⁡(X)\operatorname{Var}(3X+5)=9\operatorname{Var}(X)
Worked solution
  1. Square the scale and remove the constant shift.
  2. Establish and simplify this mathematical relation.
    Var⁡(3X+5)=9Var⁡(X)\operatorname{Var}(3X+5)=9\operatorname{Var}(X)
  3. Establish and simplify this mathematical relation.
    Var⁡(3X+5)=18\operatorname{Var}(3X+5)=18
  4. Variance has squared units.
18
Checks and common pitfalls: Variance has squared units.
Think first. Reveal a hint when the class is ready.
05 / Foundation#Your turn
Two draws from a finite bag are made without replacement. Which exact count model is appropriate?
  • Draw uniformly from the remaining finite bag without replacement, except the explicitly compared replacement model.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. Binomial without checking conditions
  2. Hypergeometric
  3. Normal exactly
  4. Always uniform over counts
Working and explanation
BUILD THE REASONING
Hint 1
Track the finite population without replacement.
Hint 2
Start from this intermediate relation.
P(X=k)=(Kk)(N−Kn−k)/(Nn)P(X=k)=\binom Kk\binom{N-K}{n-k}/\binom Nn
Worked solution
  1. Track the finite population without replacement.
  2. Establish and simplify this mathematical relation.
    P(X=k)=(Kk)(N−Kn−k)/(Nn)P(X=k)=\binom Kk\binom{N-K}{n-k}/\binom Nn
  3. Establish and simplify this mathematical relation.
    max⁡(0,n−N+K)≤k≤min⁡(n,K)\max(0,n-N+K)\le k\le\min(n,K)
  4. Individual samples may be equally likely while success counts are not.
Hypergeometric
Checks and common pitfalls: Individual samples may be equally likely while success counts are not.
Think first. Reveal a hint when the class is ready.
06 / Foundation#Your turn
X~N(5,9). What is its standard deviation?
X∼N(5,9)X\sim N(5,9)
  • The normal model has positive standard deviation; its second parameter is variance.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. 9
  2. 81
  3. 3
  4. 5
Working and explanation
BUILD THE REASONING
Hint 1
Read the second parameter as variance.
Hint 2
Start from this intermediate relation.
σ2=9\sigma^2=9
Worked solution
  1. Read the second parameter as variance.
  2. Establish and simplify this mathematical relation.
    σ2=9\sigma^2=9
  3. Establish and simplify this mathematical relation.
    σ=3\sigma=3
  4. Standard deviation is the positive square root.
3
Checks and common pitfalls: Standard deviation is the positive square root.
Think first. Reveal a hint when the class is ready.
07 / Foundation#Your turn
Observed y=8 and predicted ŷ=6. What is the residual?
y=8; y^=6y=8;\ \hat y=6
  • Keep the observed pairs; unique slope needs Sxx>0, correlation also Syy>0, and causal claims need a suitable design.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. −2
  2. 14
  3. 2
  4. 6
Working and explanation
BUILD THE REASONING
Hint 1
Subtract predicted from observed.
Hint 2
Start from this intermediate relation.
e=y−y^e=y-\hat y
Worked solution
  1. Subtract predicted from observed.
  2. Establish and simplify this mathematical relation.
    e=y−y^e=y-\hat y
  3. Establish and simplify this mathematical relation.
    e=2e=2
  4. The sign records which side of the fitted line the observation lies on.
2
Checks and common pitfalls: The sign records which side of the fitted line the observation lies on.
Think first. Reveal a hint when the class is ready.
08 / Foundation#Your turn
Can r=0 rule out every relationship between x and y?
r=0r=0
  • Keep the observed pairs; unique slope needs Sxx>0, correlation also Syy>0, and causal claims need a suitable design.
Skills and prerequisite lessons
Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.
  1. Yes, all variables are independent.
  2. No; a nonlinear relationship can remain.
  3. Yes, all y-values are zero.
  4. No; r must be between 0 and 1 only.
Working and explanation
BUILD THE REASONING
Hint 1
Consider a symmetric quadratic scatter.
Hint 2
Start from this intermediate relation.
x=(−1,0,1); y=(1,0,1)x=(-1,0,1);\ y=(1,0,1)
Worked solution
  1. Consider a symmetric quadratic scatter.
  2. Establish and simplify this mathematical relation.
    x=(−1,0,1); y=(1,0,1)x=(-1,0,1);\ y=(1,0,1)
  3. Establish and simplify this mathematical relation.
    Sxy=0; y=x2S_{xy}=0;\ y=x^2
  4. Pearson r measures linear association.
No; a nonlinear relationship can remain.
Checks and common pitfalls: Pearson r measures linear association.
Think first. Reveal a hint when the class is ready.
Focus on one question
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