Discrete vs Continuous Probability
The most fundamental distinction in probability theory: is your variable discrete (countable outcomes) or continuous (infinite, uncountable outcomes)?
Discrete random variables take specific, countable values: number of heads, dice roll outcomes, students in a class. Continuous random variables take any value in a range: height, time, temperature, weight.
| Feature | Discrete | Continuous |
|---|---|---|
| Values | Countable (0, 1, 2, …) | Any real value in a range |
| Probability of exact value | P(X = x) can be > 0 | P(X = x) = 0 always |
| Probability function | PMF — P(X = x) | PDF — f(x) |
| Sum/integral = 1 | Σ P(xᵢ) = 1 | ∫ f(x) dx = 1 |
| P(a ≤ X ≤ b) | Sum PMF values | Integrate PDF |
| Expected value | E(X) = Σ x·P(x) | E(X) = ∫ x·f(x) dx |
| Key distributions | Binomial, Poisson, Geometric | Normal, Uniform, Exponential |
| Examples | Dice roll, coin flips, defects | Height, weight, time, temperature |
Discrete Probability in Depth
A probability mass function (PMF) assigns a probability to each possible value: P(X = x) ≥ 0, and all probabilities sum to 1.
Key discrete distributions: Bernoulli (single yes/no trial), Binomial (k successes in n trials), Poisson (count of rare events), Geometric (trials until first success), Hypergeometric (sampling without replacement).
Worked example: Rolling a d6. PMF: P(X = 1) = P(X = 2) = … = P(X = 6) = 1/6. P(X ≤ 3) = 3/6 = 0.5.
Continuous Probability in Depth
A probability density function (PDF) f(x) ≥ 0 with ∫f(x)dx = 1. Probability of a range: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
P(X = any specific value) = 0 for continuous variables. Instead, we always state probabilities as intervals: "P(height between 170 and 175 cm)."
Key continuous distributions: Normal/Gaussian (bell curve), Uniform (equal density), Exponential (time between events), Beta (probabilities), Gamma (waiting times).
Worked example: Standard Normal (mean=0, sd=1). P(−1 ≤ Z ≤ 1) ≈ 68.27% (empirical rule).
How to Decide Which Applies
Ask: Can the variable take every value in some interval, or only isolated values? Countable → discrete. Uncountable → continuous.
Practical rule: money (rounded to cents) is technically discrete but often modelled as continuous for simplicity. Time and physical measurements are almost always continuous. Counts are almost always discrete.
Verdict
Use discrete probability for counts and specific outcomes; use continuous probability for measurements that can take any value in a range.
- ✓Discrete: when outcomes are countable — dice, cards, coin flips, defect counts, goals in a match.
- ✓Continuous: when outcomes are measurements — height, weight, time, temperature, returns.
- ✓The mathematical tools differ: PMF vs PDF, summation vs integration — but the conceptual framework is identical.
- ✓Many real problems mix both: a zero-inflated distribution is discrete at zero and continuous elsewhere.