Section 07: Probability & Statistics
This exam covers all material from Sections 01-07, with emphasis on probability!
| 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 | \(P(A\|B) = \frac{P(B\|A) \cdot P(A)}{P(B)}\) |
| 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 | \(E[aX+b]=aE[X]+b\), \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\) |
| 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 | \(P(D'\|-)\) |
| 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 | \(\int_a^b [f(x) - g(x)]dx\) |
No communication with other students during the exam!
| Problem | Topic | Points |
|---|---|---|
| Problem 1 | Functions and Calculus | 30 |
| Problem 2 | Integration and Applications | 30 |
| Problem 3 | Probability and Statistics | 40 |
| Total | 100 |
See exam handout for full problem.
Topics covered: - Function analysis (domain, zeros, extrema) - Derivative calculations - Curve sketching - Tangent line equations
See exam handout for full problem.
Topics covered: - Indefinite integrals - Integration by parts - Definite integrals - Area between curves - Business applications
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)
Key Points
Integration by Parts Reminder
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: \(-e^{-x}(x^2 + 2x + 2) + C\)
Contingency Table Strategy
Bayes’ Theorem Strategy
\[P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+|D) \cdot P(D) + P(+|D') \cdot P(D')}\]
Or use the contingency table method with a hypothetical population!
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.
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 |
Based on your performance, prioritize:
Choose your track based on score and error profile.
Study Strategy
Use Section 07 diagnostics to plan Section 08 and 09 work.
The probability section is now complete. Make sure you’re confident with all material before the final exam!
Session 07-08 - Mock Exam 2 | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home