Stats 100A MT Review v0

In probability theory, what is the definition of the sample space Ω?

The set of all possible outcomes of a random experiment.

What is the mathematical definition of an event A in a random experiment?

A subset of the sample space Ω (A ⊆ Ω).

Under Axiom 0 (classical probability), what is the formula for P(A) when all outcomes in Ω are equally likely?

P(A) = |A| / |Ω|

In a population framework, how is the probability of an event interpreted?

As the population proportion of individuals or outcomes satisfying that event.

What is the mathematical formula for conditional probability P(A | B), given P(B) > 0?

P(A | B) = P(A ∩ B) / P(B)

How can unconditioned probability P(A) be expressed as a conditional probability?

P(A) = P(A | Ω)

What is the two-event chain rule for P(A ∩ B)?

P(A ∩ B) = P(A)P(B | A) = P(B)P(A | B)

What is the chain rule formula for three events P(A ∩ B ∩ C)?

P(A ∩ B ∩ C) = P(A)P(B | A)P(C | A ∩ B)

What is the general chain rule formula for n events P(A₁ ∩ A₂ ∩ … ∩ Aₙ)?

P(A₁ ∩ A₂ ∩ … ∩ Aₙ) = P(A₁)P(A₂ | A₁)P(A₃ | A₁ ∩ A₂) … P(Aₙ | A₁ ∩ … ∩ Aₙ₋₁)

What is the formula for the Law of Total Probability for a partition B₁, B₂, …, Bₖ?

P(A) = Σᵢ₌₁ᵏ P(Bᵢ)P(A | Bᵢ)

What two properties must sets B₁, B₂, …, Bₖ satisfy to form a partition of the sample space Ω?

1. Mutually exclusive: Bᵢ ∩ Bⱼ = ∅ for i ≠ j; 2. Exhaustive: ⋃ᵢ₌₁ᵏ Bᵢ = Ω.

What is Bayes' rule for calculating P(A | B) using P(B | A) and P(A)?

P(A | B) = P(A)P(B | A) / P(B) = [P(A)P(B | A)] / [P(A)P(B | A) + P(Aᶜ)P(B | Aᶜ)]

In Bayesian inference, what is the term for P(A) versus P(A | B)?

P(A) is the prior probability, and P(A | B) is the posterior probability.

What cognitive error occurs when conditional probability P(A | B) is evaluated while ignoring the low overall prior probability P(A)?

Base rate neglect.

What is the definition of independence for two events A and B?

P(A ∩ B) = P(A)P(B).

If P(B) > 0, how is event independence A ⟂ B expressed in terms of conditional probability?

P(A | B) = P(A).

If two non-zero probability events A and B are disjoint (A ∩ B = ∅), can they be independent?

No. Disjoint events with non-zero probabilities are mutually exclusive and cannot be independent because P(A ∩ B) = 0 ≠ P(A)P(B).

What condition defines pairwise independence for three events A, B, and C?

P(A ∩ B) = P(A)P(B); P(A ∩ C) = P(A)P(C); P(B ∩ C) = P(B)P(C).

What four equations must hold for three events A, B, and C to be mutually independent?

1. P(A ∩ B) = P(A)P(B); 2. P(A ∩ C) = P(A)P(C); 3. P(B ∩ C) = P(B)P(C); 4. P(A ∩ B ∩ C) = P(A)P(B)P(C).

Does pairwise independence of three events imply mutual independence?

No. Pairwise independence does not imply mutual independence.

What is the mathematical definition of conditional independence A ⟂ C | B?

P(A ∩ C | B) = P(A | B)P(C | B), or equivalently P(A | B ∩ C) = P(A | B).

According to the multiplication principle, if step 1 has n₁ outcomes and step 2 has n₂ outcomes, how many total combined outcomes exist?

n₁ × n₂.

How many total ordered sequences of length k can be formed from n elements when sampling with replacement?

nᵏ.

What is the permutation formula P(n,k) for selecting k ordered items from n distinct items without replacement?

P(n,k) = n! / (n-k)!

What is the combination formula C(n,k) for selecting k unordered items from n distinct items without replacement?

C(n,k) = n! / [k!(n-k)!]

What symmetry identity holds for combinations C(n,k)?

C(n,k) = C(n,n-k).

What is Pascal's rule for combinations?

C(n,k) = C(n-1,k-1) + C(n-1,k).

What is the Binomial Theorem formula for expanding (a + b)ⁿ?

(a + b)ⁿ = Σₖ₌₀ⁿ C(n,k)aᵏbⁿ⁻ᵏ.

In combinatorics, which selection formula is used when sampling without replacement and order does NOT matter?

Combination: C(n,k) = n! / [k!(n-k)!].

In combinatorics, which selection formula is used when sampling without replacement and order DOES matter?

Permutation: P(n,k) = n! / (n-k)!.

What is a discrete random variable?

A function X: Ω → ℝ that maps elements of a sample space to a countable set of numerical values.

What is the Probability Mass Function (PMF) p(x) of a discrete random variable X?

p(x) = P(X = x).

What are the two mandatory properties required for any function p(x) to be a valid PMF(Probability of Mass Function)?

1. Non-negativity: p(x) ≥ 0 for all x; 2. Normalization: Σₓ p(x) = 1.

What is the formula for the expected value E(X) of a discrete random variable X with PMF p(x)?

E(X) = Σₓ x · p(x).

What is the formula for the Law of the Unconscious Statistician (LOTUS) to compute E(h(X)) for a discrete random variable X?

E(h(X)) = Σₓ h(x)p(x).

What is the linearity of expectation property for E(aX + b), where a and b are constants?

E(aX + b) = aE(X) + b.

What is the definition of variance Var(X) in terms of expectation?

Var(X) = E[(X - E(X))²].

What is the computational shortcut formula for variance Var(X)?

Var(X) = E(X²) - [E(X)]².

How does adding a constant c affect the variance of a random variable Var(X + c)?

Var(X + c) = Var(X).

How does multiplying a random variable by a constant c affect its variance Var(cX)?

Var(cX) = c²Var(X).

What models a single trial with two outcomes (success with probability p, failure with probability 1-p)?

Bernoulli distribution: X ~ Bernoulli(p).

What are the expected value E(X) and variance Var(X) of X ~ Bernoulli(p)?

E(X) = p; Var(X) = p(1-p).

What distribution represents the sum of n independent and identically distributed Bernoulli(p) trials?

Binomial distribution: X ~ Binomial(n,p).

What is the PMF formula for X ~ Binomial(n,p)?

P(X = k) = C(n,k)pᵏ(1-p)ⁿ⁻ᵏ for k = 0, 1, …, n.

What are the expected value E(X) and variance Var(X) of X ~ Binomial(n,p)?

E(X) = np; Var(X) = np(1-p).

In a basketball streak analysis, why does selecting sequences that already contain a streak of k successes bias the conditional probability of the next shot downward?

Conditioning on past values within finite samples introduces selection bias, making the next outcome appear less likely than p even for independent fair coin flips.

What distribution counts the number of events occurring in a fixed interval of time or space with rate parameter λ?

Poisson distribution: X ~ Poisson(λ).

What is the PMF formula for X ~ Poisson(λ)?

P(X = k) = (λᵏe⁻λ) / k! for k = 0, 1, 2, …

What are the expected value E(X) and variance Var(X) of X ~ Poisson(λ)?

E(X) = λ; Var(X) = λ.

If X₁ ~ Poisson(λ₁) and X₂ ~ Poisson(λ₂) are independent, what is the distribution of X₁ + X₂?

X₁ + X₂ ~ Poisson(λ₁ + λ₂).

What distribution models the number of independent Bernoulli trials needed to achieve the first success?

Geometric distribution.

What is the PMF formula for X ~ Geometric(p), where X is the trial number of the first success?

P(X = k) = (1-p)ᵏ⁻¹p for k = 1, 2, 3, …

For three fair coins thrown simultaneously, what is the probability that all three show the same face on a single throw?

2/8 = 1/4 (HHH or TTT out of 8 outcomes).

Three fair coins are thrown until all three show the same face (probability of success p = 1/4 per throw). What is the probability that they are thrown more than 3 times?

(1 - 1/4)³ = (3/4)³ = 27/64.

If (X,Y) is chosen uniformly from the unit square [0,1]², what is P(X² + Y² ≤ 1)?

Area of quarter circle / Area of square = (π/4)/1 = π/4.

If m out of n uniform random points in the unit square fall within X² + Y² ≤ 1, how is π approximated?

π ≈ 4(m/n).

For independent uniform random variables X,Y ~ Uniform[0,1], what is P(X ≥ 1/2 | X + Y ≥ 1)?

3/4.

If player A wins a set against player B with probability 2/3, what is the probability player A wins a best-of-5 match (first to win 3 sets)?

192/243 ≈ 0.7901.

Five bowls are labeled i = 1, 2, 3, 4, 5. Bowl i has i white and 5-i black balls. If a bowl is picked at random and 2 balls are drawn without replacement, what is P(both white)?

(1/5)Σᵢ₌₁⁵ [C(i,2)/C(5,2)] = (1/50)(0 + 1 + 3 + 6 + 10) = 20/50 = 2/5.

What is the formula for the probability that exactly one of three arbitrary events A, B, and C occurs?

P(A) + P(B) + P(C) - 2P(A ∩ B) - 2P(A ∩ C) - 2P(B ∩ C) + 3P(A ∩ B ∩ C).

In a two-dice game, each player scores the difference between the larger and smaller numbers. How many total outcomes are in the sample space for one roll of two six-sided dice?

36 outcomes.

When rolling 3 fair six-sided dice independently, what is the probability that all 3 show different numbers?

(6 × 5 × 4) / 6³ = 120/216 = 5/9.

When rolling 3 fair six-sided dice independently, what is the probability that at least one die shows a 6?

1 - (5/6)³ = 1 - 125/216 = 91/216.

Let spinner outcome X ∈ {1,2,3,4,5} with equal probability 1/5. If prize payout is h(X) = X² - 5 dollars, what entry fee c makes the game fair (E[h(X) - c] = 0)?

c = E(h(X)) = E(X²) - 5 = 11 - 5 = 6 dollars.

If calls arrive according to a Poisson process with rate λ = 4 calls/hour, what is the probability of receiving zero calls in a given hour?

P(X = 0) = (4⁰e⁻⁴)/0! = e⁻⁴.

A call center receives calls with P(zero calls in 1 hour) = e⁻⁴. Over 20 independent hours, let Y be the number of hours with zero calls. What is the distribution and expected value of Y?

Y ~ Binomial(20,e⁻⁴), with E(Y) = 20e⁻⁴.

According to the Law of Large Numbers (LLN), what happens to the sample average (X₁ + … + Xₙ)/n of independent trials as n → ∞?

It converges to the population expected value E(X).

What does choosing positions (1,3) or (3,1) to obtain the same coin sequence HTHT illustrate about combinations versus permutations?

Choosing positions (1,3) or (3,1) yields the exact same outcome HTHT, demonstrating that order of position selection does not matter for combinations.

In a discrete random variable mapping X(ω): Ω → {1,2,3,4} for eye color, how is P(X = x) computed from counts?

P(X = x) = N(x)/N, where N(x) is the count of individuals with feature value x and N is the total population size.

Using a histogram of sibling counts from 0 to 6, how do you verify that the empirical PMF is normalized?

Sum the counts for all categories x ∈ {0,1,…,6} and divide by total N; the resulting sum of probabilities must equal 1.

What is the rule for computing the probability of the complement event Aᶜ?

P(Aᶜ) = 1 - P(A).

What is the general inclusion-exclusion principle for two events A and B?

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

If A and B are independent, what is P(A ∪ B) in terms of P(A) and P(B)?

P(A ∪ B) = P(A) + P(B) - P(A)P(B).

What is the multinomial coefficient formula for dividing n items into groups of sizes k₁, k₂, …, kᵣ where Σkᵢ = n?

C(n; k₁,k₂,…,kᵣ) = n! / (k₁!k₂!…kᵣ!).

In a population with N total people, N₁ males, N₀ females, T₁ tall males, and T₀ tall females, what is P(Male | Tall)?

P(Male | Tall) = T₁ / (T₁ + T₀).

How do you distinguish between a Law of Total Probability problem and a Bayes' Rule problem when given partition probabilities P(B_i) and conditional probabilities P(A|B_i)?

Law of Total Probability computes the unconditioned overall probability P(A), whereas Bayes' Rule computes the updated conditional probability of a specific partition given the observed outcome, P(B_k|A).

In a word problem asking 'Given that event A occurred, what is the probability that it came from cause B_1?', what is the exact initial formula to write down?

Write Bayes' Rule: P(B_1|A) = (P(B_1)P(A|B_1))/(P(A)) = (P(B_1)P(A|B_1))/(Σ_i P(B_i)P(A|B_i)).

When a probability word problem asks for the probability of 'at least one success' in n independent trials, what is the standard first mathematical move?

Apply the Complement Rule: P(at least one success) = 1 - P(zero successes) = 1 - (1 - p)ⁿ.

In a counting problem, how do you decide whether to use a permutation P_n,k or a combination C(n, k)?

Use permutations P_n,k if the ordering or arrangement of selected items creates distinct outcomes; use combinations C(n, k) if choosing an unordered subset.

What formula counts the total number of distinct sequences when choosing k items from n distinct options with replacement when order matters?

nᵏ total possible sequences.

In a game of chance with payout function h(X) and entry fee c, what condition defines a 'fair game'?

A game is fair if expected net profit is zero: E[h(X) - c] = 0, which implies c = E[h(X)].

Given a discrete PMF p(x) and a payout function h(x), what formula allows you to calculate expected payout E[h(X)] directly?

Use the Law of the Unconscious Statistician (LOTUS): E[h(X)] = Σ_x h(x) p(x).

What four essential conditions signal that a random variable X follows a Binomial(n, p) distribution?

1. Fixed number of trials n 2. Two binary outcomes per trial (success/failure) 3. Constant success probability p 4. Independent trials

If X sim Binomial(n, p), what is the PMF formula used to compute P(X = k)?

P(X = k) = C(n, k) pᵏ (1 - p)ⁿ ⁻ ᵏ for k ∈ 0, 1, dots, n.

What key trigger phrasing indicates a random variable T follows a Geometric distribution, and what is its expected value E(T)?

Trigger phrasing: 'Number of independent trials until the first success.' Expectation: E(T) = (1)/(p).

What key problem phrasing signals a Poisson distribution with parameter lambda, and what is its PMF for X = k?

Trigger phrasing: 'Events occur randomly at a constant average rate lambda over a fixed time interval.' PMF: P(X = k) = (e⁻ˡᵃᵐᵇᵈᵃ lambdaᵏ)/(k!) for k = 0, 1, 2, dots.

Given a random variable X with variance Var(X) and constants a, b, what is Var(aX + b)?

Var(aX + b) = a² Var(X). (The constant shift b does not change the variance).

Given a random variable X and constants a, b, what is E(aX + b)?

E(aX + b) = a E(X) + b, by the linearity of expectation.

Diagram depicting a large group of people labelled Population and a smaller subgroup labelled Sample What statistical distinction does this visual represent between population parameters and sample statistics?

A population represents the entire set of interest (with true mean mu), whereas a sample is a subset randomly drawn to estimate population parameters via sample statistics like barx = (1)/(n) Σ x_i.

Galton board path probabilities on the left and Pascal triangle numbers on the right How does a 5-layer Galton board determine the probability of a ball landing in bin k (k ∈ 0, 1, 2, 3, 4, 5)?

Each layer split represents an independent Bernoulli trial with p = 0.5. The probability of landing in bin k after 5 steps is given by Binomial(5, 0.5): P(X = k) = C(5, k) (0.5)⁵ = (C(5, k))/(32).

What is the shortcut computational formula for Var(X) in terms of raw moments?

Var(X) = E(X²) - [E(X)]².

How is the Standard Deviation SD(X) defined in terms of Variance Var(X), and what are its units?

SD(X) = sqrt(Var(X)). Its units match the original units of X (whereas variance is in squared units).

Given a random variable X with mean mu and variance sigma², how is the standardized variable Z defined, and what are its mean and variance?

Z = (X - mu)/(sigma). It has E(Z) = 0 and Var(Z) = 1.

How is the chain rule used to compute the joint probability P(A ∩ B) using conditional probability?

P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B).

If two non-impossible events A and B are mutually exclusive (A ∩ B = ∅), can they be independent?

No. Mutually exclusive non-impossible events cannot be independent because P(A ∩ B) = 0 ≠ P(A)P(B).

What is the formula definition for the conditional probability P(A|B)?

P(A|B) = (P(A ∩ B))/(P(B)), provided P(B) > 0.

What exact condition proves that two events A and B are statistically independent?

P(A ∩ B) = P(A) P(B), or equivalently P(A|B) = P(A) when P(B) > 0.

If X sim Binomial(n, p), how can X be represented as a sum of Bernoulli indicator random variables?

X = Σ_i=1ⁿ Z_i, where each Z_i sim Bernoulli(p) independently, with Z_i = 1 for success and Z_i = 0 for failure.

If Z_1, Z_2, dots, Z_n are independent random variables, what is Var(Σ_i=1ⁿ Z_i)?

Var(Σ_i=1ⁿ Z_i) = Σ_i=1ⁿ Var(Z_i). (Independence is required for variances to add linearly).

If X sim Binomial(n, p) and sample proportion is hatp = (X)/(n), what are E(hatp) and Var(hatp)?

E(hatp) = p and Var(hatp) = (p(1 - p))/(n).