Probability distributions and simulations - Mathematics: Applications & Interpretation IB Study Notes

Probability distributions describe how probabilities are distributed over the possible values of a random variable. A random variable is a numerical outcome of a random process, denoted by capital letters (X, Y) with specific values in lowercase (x, y).

Discrete probability distributions assign probabilities to countable outcomes. The probability mass function P(X = x) must satisfy: (1) 0 ≤ P(X = x) ≤ 1 for all x, and (2) ΣP(X = x) = 1. The expected value (mean) is E(X) = μ = Σx·P(X = x), representing the long-run average. Variance measures spread: Var(X) = σ² = E(X²) - [E(X)]² = Σx²·P(X = x) - μ².

Continuous probability distributions use a probability density function f(x) where probabilities are areas under the curve: P(a ≤ X ≤ b) = ∫ᵃᵇf(x)dx. For continuous distributions, P(X = x) = 0 for any specific value.

The binomial distribution B(n, p) models n independent trials with constant success probability p: P(X = r) = ⁿCᵣ·pʳ·(1-p)ⁿ⁻ʳ, with E(X) = np and Var(X) = np(1-p).

The normal distribution N(μ, σ²) is symmetric and bell-shaped, defined by mean μ and variance σ². The standard normal Z ~ N(0, 1) uses the transformation Z = (X - μ)/σ.

Simulations use technology to model random processes by generating random numbers to estimate probabilities and expected values, particularly useful when theoretical calculations are complex. They require: defining the model, generating random data, repeating trials, and analyzing results.

Think of probability distributions as blueprints for randomness—they tell you how likely different outcomes are before they happen. Just as a weather forecast gives percentage chances for rain, probability distributions quantify uncertainty mathematically.

Real-world binomial applications: Quality control in manufacturing uses binomial distributions. If 5% of light bulbs are defective (p = 0.05), testing 20 bulbs (n = 20) follows B(20, 0.05). A pharmaceutical company testing if exactly 2 out of 20 patients experience side effects uses P(X = 2) = ²⁰C₂(0.05)²(0.95)¹⁸ ≈ 0.189. Expected defects: E(X) = 20(0.05) = 1 bulb.

Normal distribution examples: Human heights, IQ scores, and measurement errors follow normal distributions. If heights are N(170, 64) cm, the probability someone is between 165-175 cm involves standardizing: Z₁ = (165-170)/8 = -0.625 and Z₂ = (175-170)/8 = 0.625, then using tables or GDC to find P(-0.625 < Z < 0.625) ≈ 0.468.

Simulations in practice: Insurance companies simulate thousands of accident scenarios to price premiums. A Monte Carlo simulation might generate 10,000 random claim amounts to estimate expected annual payouts. Climate scientists simulate weather patterns; epidemiologists model disease spread using random infection probabilities.

Memory aid: BINOMIAL = Binary outcomes, Independent trials, Number fixed, Only two outcomes, Maintained probability, Identical conditions, Analyzing discrete events, Long-run predictions.

Simulations bridge theory and reality—when formulas become unwieldy, computers can run millions of trials to approximate answers that would take hours by hand.