Session 07-08 - Mock Exam 2
Section 07: Probability & Statistics
Mock Exam 2 - Overview
Today’s Session Structure
- Part 1: Review of key formulas (15 minutes)
- Part 2: Mock Exam (120 minutes)
- Part 3: Break (15 minutes)
- Part 4: Solution discussion (30 minutes)
. . .
This exam covers all material from Sections 01-07, with emphasis on probability!
Formula Review
Probability Formulas
| Concept | Formula |
|---|---|
| Complement | \(P(A') = 1 - P(A)\) |
| Addition | \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) |
| Conditional | \(P(A\|B) = \frac{P(A \cap B)}{P(B)}\) |
| Multiplication | \(P(A \cap B) = P(A\|B) \cdot P(B)\) |
| Independence | \(P(A \cap B) = P(A) \cdot P(B)\) |
| Bayes | \(P(A\|B) = \frac{P(B\|A) \cdot P(A)}{P(B)}\) |
Counting and Distributions
| Concept | Formula |
|---|---|
| Permutation | \(P(n,r) = \frac{n!}{(n-r)!}\) |
| Combination | \(C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\) |
| Hypergeometric | \(P(X=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}\) |
| Binomial | \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\) |
| Binomial mean | \(\mu = np\) |
| Binomial std | \(\sigma = \sqrt{np(1-p)}\) |
| Geometric | \(P(X=n) = (1-p)^{n-1}p\) |
| Geometric cumulative | \(P(X\le n)=1-(1-p)^n\) |
| Z-score | \(Z=\frac{X-\mu}{\sigma}\) |
| Linear transformations | \(E[aX+b]=aE[X]+b\), \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\) |
Medical Testing
| Metric | Definition |
|---|---|
| Sensitivity | \(P(+\|D)\) |
| Specificity | \(P(-\|D')\) |
| False positive rate | \(P(+\|D') = 1 - \text{Specificity}\) |
| False negative rate | \(P(-\|D) = 1 - \text{Sensitivity}\) |
| PPV | \(P(D\|+)\) |
| NPV | \(P(D'\|-)\) |
Calculus Review
| Concept | Formula |
|---|---|
| Power rule (diff) | \((x^n)' = nx^{n-1}\) |
| Power rule (int) | \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) |
| Integration by parts | \(\int u\,dv = uv - \int v\,du\) |
| Definite integral | \(\int_a^b f(x)dx = F(b) - F(a)\) |
| Area between curves | \(\int_a^b [f(x) - g(x)]dx\) |
Mock Exam Instructions
Exam Rules
- Duration: 120 minutes
- Materials allowed: Calculator, formula sheet
- Show all work: Partial credit is awarded
- Answer ALL questions
- Box your final answers
. . .
No communication with other students during the exam!
Grading Breakdown
| Problem | Topic | Points |
|---|---|---|
| Problem 1 | Functions and Calculus | 30 |
| Problem 2 | Integration and Applications | 30 |
| Problem 3 | Probability and Statistics | 40 |
| Total | 100 |
Begin Mock Exam
Problem 1: Functions and Calculus (30 points)
See exam handout for full problem.
Topics covered: - Function analysis (domain, zeros, extrema) - Derivative calculations - Curve sketching - Tangent line equations
Problem 2: Integration (30 points)
See exam handout for full problem.
Topics covered: - Indefinite integrals - Integration by parts - Definite integrals - Area between curves - Business applications
Problem 3: Probability (40 points)
See exam handout for full problem.
Topics covered: - Contingency tables - Conditional probability - Bayes’ theorem - Distribution selection (binomial vs hypergeometric vs geometric) - Binomial distribution - Normal probabilities with z-scores - Medical testing (sensitivity, specificity, PPV)
Break - 15 Minutes
Solution Discussion
Problem 1 Solutions
- Always check domain first
- Use first and second derivative tests systematically
- Verify extrema and inflection points
- Tangent line: \(y - f(a) = f'(a)(x - a)\)
Problem 2 Solutions
For \(\int xe^x dx\): - Choose \(u = x\) (algebraic), \(dv = e^x dx\) - Result: \(e^x(x-1) + C\)
For \(\int x^2 e^{-x} dx\): - Apply twice - Result: \(-e^{-x}(x^2 + 2x + 2) + C\)
Problem 3 Solutions
- Draw the table first
- Fill in given values
- Use row/column sums to find unknowns
- Calculate probabilities directly from table
\[P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+|D) \cdot P(D) + P(+|D') \cdot P(D')}\]
Or use the contingency table method with a hypothetical population!
Error Taxonomy (Use This After Every Mock)
Classify each lost point before re-solving.
| Error Type | Typical Sign | Fix Strategy |
|---|---|---|
| Concept error | Wrong method family chosen | Revisit concept card + 2 basic examples |
| Setup error | Correct idea, wrong equation/table setup | Rewrite givens in notation before calculating |
| Algebra/arithmetic error | Formula right, computation wrong | Slow down and add line-by-line checks |
| Interpretation error | Numeric answer but wrong meaning | Add one sentence in plain business language |
Do not just “redo everything”. First identify your error type, then target the correction.
Self-Assessment
Evaluate Your Performance
Score yourself honestly:
| Score Range | Assessment |
|---|---|
| 85-100 | Excellent - well prepared |
| 70-84 | Good - minor review needed |
| 55-69 | Satisfactory - focused review needed |
| Below 55 | Needs work - comprehensive review |
Areas for Review
Based on your performance, prioritize:
- Integration by parts (if missed Problem 2c)
- Contingency tables (if missed Problem 3a)
- Bayes’ theorem (if missed Problem 3b)
- Binomial distribution (if missed Problem 3c)
Remediation Tracks
Choose your track based on score and error profile.
- Track A (85-100): Maintain speed and precision
- One timed mixed set per day
- Focus on avoiding careless mistakes
- Track B (70-84): Close specific topic gaps
- Two targeted sets on weakest topic
- One mixed review set every second day
- Track C (55-69): Rebuild core methods
- Rework foundational examples step by step
- Daily short quiz on notation and setup
- Track D (<55): Structured recovery plan
- Start with concept summaries + worked examples
- Move to guided problems before timed practice
Preparation for Final Exam
Remaining Sessions
- Section 08: Financial Mathematics (2 sessions)
- Section 09: Synthesis & Final Preparation (5 sessions)
- Section 10: Final Confidence Session
Key Recommendations
- Review all mock exams and understand every solution
- Practice integration by parts until automatic
- Create a comprehensive formula sheet
- Time yourself on practice problems
- Focus on your weakest areas
Bridge to Final Exam Preparation
Use Section 07 diagnostics to plan Section 08 and 09 work.
- Carry over your top 2 error categories into your weekly study plan.
- Pair each weak probability topic with one calculus/financial topic to keep skills balanced.
- Reattempt one chained exam-style problem per week (table/Bayes + binomial follow-up).
Homework Assignment
Before Next Session
- Review all solutions from today’s mock exam
- Redo any problems you didn’t solve correctly
- Create a study plan for remaining weaknesses
- Complete any unfinished tasks from Section 07
. . .
The probability section is now complete. Make sure you’re confident with all material before the final exam!