Rucete ✏ AP Statistics In a Nutshell
4. Probability, Random Variables, and Probability Distributions
This chapter introduces basic probability rules, calculations for random variables, and probability distributions including binomial and geometric distributions, as well as concepts like expected value, variance, and standard deviation.
Introduction to Probability
• Probability measures the likelihood of an event occurring, ranging between 0 and 1.
• Law of Large Numbers: As the number of trials increases, the relative frequency of an event approaches its true probability.
• Short-term irregularities (gambler’s fallacy) are not governed by probability laws.
Basic Probability Rules
• Complement Rule: P(Aᶜ) = 1 − P(A).
• Addition Rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B).
• General Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Multiplication Rule for independent events: P(A ∩ B) = P(A)P(B).
• Conditional Probability: P(A|B) = P(A ∩ B) / P(B).
• Independence: Events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B).
Understanding Independence and Mutual Exclusivity
• Mutually exclusive events cannot happen simultaneously (P(A ∩ B) = 0).
• Independent events do not influence each other’s occurrence.
• Mutually exclusive events are never independent unless one has probability 0.
Multistage Probability and Conditional Probability
• Use tree diagrams to represent conditional probabilities across multiple stages.
• Reverse conditional probabilities can be calculated using Bayes’ theorem.
• Example: Finding probability a computer has a virus given brand and infection rates.
Random Variables
• Random Variable: Assigns a numerical value to each outcome in a sample space.
• Discrete Random Variable: Finite or countable outcomes (e.g., number of heads).
• Mean (Expected Value) of X: E(X) = Σ[x × P(x)].
• Variance of X: σ² = Σ[(x − μ)² × P(x)].
• Standard Deviation: σ = √Variance.
Combining Random Variables
• For independent random variables X and Y:
• E(X ± Y) = E(X) ± E(Y).
• Var(X ± Y) = Var(X) + Var(Y).
• Standard deviations add using square roots: SD(X ± Y) = √(σX² + σY²).
Transforming Random Variables
• Adding a constant shifts the mean but does not change spread.
• Multiplying by a constant scales both mean and spread by the absolute value of the constant.
Binomial Distributions
• Binomial Setting: Fixed number of independent trials (n), each resulting in success or failure, with constant probability p of success.
• Conditions for Binomial Distribution (BINS):
• Binary: Two outcomes (success/failure).
• Independent: Trials are independent.
• Number: Fixed number of trials n.
• Success: Constant probability p of success on each trial.
Binomial Probability Formula
• Probability of exactly k successes: P(X = k) = (n choose k) × (p^k) × (1 − p)^(n − k).
• (n choose k) = n! / (k!(n − k)!).
Mean and Standard Deviation of a Binomial Distribution
• Mean (Expected Value): μ = np.
• Standard Deviation: σ = √(np(1 − p)).
• Larger sample sizes make binomial distributions approach normality if np ≥ 10 and n(1 − p) ≥ 10 (Normal Approximation to the Binomial).
Geometric Distributions
• Geometric Setting: Repeated independent trials until the first success occurs.
• Conditions for Geometric Distribution:
• Binary outcomes.
• Independent trials.
• Constant probability of success.
• Interested in the number of trials until first success.
Geometric Probability Formula
• P(X = k) = (1 − p)^(k−1) × p.
Mean and Standard Deviation of a Geometric Distribution
• Mean (Expected Value): μ = 1/p.
• Standard Deviation: σ = √((1 − p)/p²).
Interpreting Binomial and Geometric Distributions
• Binomial: Predicting the number of successes in a fixed number of trials.
• Geometric: Predicting the trial on which the first success occurs.
Calculating Cumulative Probabilities
• For binomial or geometric settings, use cumulative probabilities to find "at most," "at least," or "more than" probabilities.
• P(X ≥ k) = 1 − P(X ≤ k−1).
• Cumulative binomial tables or technology (calculator functions) can be used to simplify cumulative probability calculations.
When to Use Normal Approximation
• Use normal approximation to the binomial distribution when np ≥ 10 and n(1 − p) ≥ 10.
• Apply continuity correction when using a normal approximation (adjust boundary by 0.5 units).
Important Warnings
• Do not assume normal approximation if conditions np ≥ 10 and n(1 − p) ≥ 10 are not satisfied.
• Do not use binomial or geometric models if trials are not independent.
In a Nutshell
Probability rules help calculate the chances of events in complex situations. Random variables model numerical outcomes and are characterized by distributions, means, and spreads. Binomial and geometric distributions handle specific repeated-trial scenarios, each with their own formulas and conditions. Proper modeling, checking assumptions, and sometimes applying normal approximations are key to valid probability calculations and interpretations.