Probability and Statistics

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Probability and Statistics — Key Facts for Kenyatta University Probability: P(E) = favourable outcomes / total outcomes; always 0 ≤ P ≤ 1 Conditional: P(A|B) = P(A ∩ B)/P(B) Independent events: P(A ∩ B) = P(A)·P(B) Mean: x̄ = Σx_i/n or Σf_ix_i/Σf_i for grouped Standard deviation: σ = √[Σ(x_i − x̄)²/n] ⚡ Exam tip: For probability, always check whether events are independent or mutually exclusive — different formulas!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Probability and Statistics — Kenyatta University Study Guide

Probability rules:

• P(A or B) = P(A) + P(B) − P(A and B)
• P(A and B) = P(A)·P(B) for independent events
• P(not A) = 1 − P(A)
• P(A or B) = P(A) + P(B) for mutually exclusive events

Bayes’ theorem: P(A|B) = P(B|A)·P(A) / P(B)

Permutations: nP_r = n!/(n−r)! — order matters

Combinations: nC_r = n!/[r!(n−r)!] — order doesn’t matter

Probability distributions:

• Binomial: P(X=r) = nC_r p^r q^{n−r}
• Mean of binomial = np
• Variance of binomial = npq

Statistics:

• Mean: x̄ = Σx_i/n
• For grouped: x̄ = Σf_ix_i/Σf_i
• Median: middle value when sorted
• Mode: most frequent value

Standard deviation: σ = √[Σ(x_i − x̄)²/n] (population) s = √[Σ(x_i − x̄)²/(n−1)] (sample)

Variance shortcut: σ² = (Σx_i²)/n − x̄²

Quartiles: Q1 = value at (n+1)/4; Q3 = value at 3(n+1)/4

Correlation: r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)²·Σ(y−ȳ)²] Ranges from −1 to +1

• Key formula: P(A|B) = P(A ∩ B)/P(B); nP_r = n!/(n−r)!; nC_r = n!/[r!(n−r)!]
• Common trap: Mutually exclusive events: P(A and B) = 0; independent: P(A|B) = P(A)
• Exam weight: 2–3 questions per exam

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Probability and Statistics — Comprehensive Kenyatta Notes

Counting principles:

• Multiplication: if task A has m ways and B has n ways, A followed by B has m×n ways
• Addition: if tasks are exclusive (either A or B), total = m + n

Complement rule: P(at least one) = 1 − P(none)

Conditional probability: P(A|B) = P(A ∩ B)/P(B) When drawing without replacement, probabilities change after each draw

Total probability: P(A) = Σ P(E_i)·P(A|E_i) for partition {E_i}

Random variable: Discrete: takes specific values with probabilities Continuous: described by probability density function

Expected value: E(X) = Σ p_i x_i E(X²) = Σ p_i x_i² Var(X) = E(X²) − [E(X)]²

Binomial distribution conditions:

1. Fixed number of trials n
2. Each trial has two outcomes (success/failure)
3. Probability p is constant
4. Trials are independent

Poisson approximation to binomial: For large n, small p: λ = np; P(X=r) ≈ e^{−λ} λ^r/r!

Normal distribution: Bell-shaped curve, symmetric about mean 68% within 1σ, 95% within 2σ, 99.7% within 3σ

Z-score: z = (x − x̄)/σ Standardises any normal distribution to N(0,1)

Hypothesis testing: Null hypothesis H₀: assumed true until evidence suggests otherwise Significance level α: probability of rejecting H₀ when true p-value: probability of observing data given H₀ is true

Regression line: y on x: y − ȳ = r(σ_y/σ_x)(x − x̄) x on y: x − x̄ = r(σ_x/σ_y)(y − ȳ)

Chi-square test: Σ(O−E)²/E for goodness of fit

Permutation with repetition: n!/(n₁!n₂!… ) for n objects with groups of identical objects

Derangements: Number of ways n objects can be arranged with no object in original position: D_n = n![1 − 1/1! + 1/2! − … + (−1)^n/n!]

Probability with dice/cards:

• P(sum of two dice = 7) = 6/36 = 1/6
• P(drawing ace from pack) = 4/52 = 1/13

Odds:

• Odds in favour of E: P(E)/P(E’) = a:b → P(E) = a/(a+b)

• Remember: P(A|B) = P(A ∩ B)/P(B); independent: P(A∩B) = P(A)·P(B); mutually exclusive: P(A∪B) = P(A)+P(B); complement: P(at least one) = 1 − P(none)

• Previous years: “Find probability of drawing 2 aces from pack without replacement” [2023 KU]; “Find mean and variance of binomial with n=10, p=0.5” [2024 KU]; “Calculate standard deviation of data: 2, 4, 6, 8, 10” [2024 KU]

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📊 Kenyatta University Exam Essentials

Detail	Value
Questions	50 (depending on course)
Time	2–3 hours
Marks	100
Format	Mix of short answer and problem solving

💡 Pro Tips

• For “at least one” problems, use complement: 1 − P(none)
• When drawing without replacement, multiply sequentially with updated counts
• For independent events, P(A∩B) = P(A)·P(B); for mutually exclusive, P(A∪B) = P(A)+P(B)
• Always check whether order matters — permutations vs combinations
• Variance shortcut: σ² = (Σx²)/n − x̄² saves computation time

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