Probability Study Guide: 102 Flashcards on Rules, Distributions and Paradoxes

Probability is the one branch of maths where intuition is actively wrong: the gambler waiting for a "due" red, the 23 people who probably share a birthday, the Monty Hall door you should always switch. Flip It deck EX3246, Probability: Rules & Distributions, puts 102 cards between you and those mistakes — a first-course syllabus followed by a tail of paradoxes.

Eight cards from the deck

What the 102 cards cover

• Foundations — probability between 0 and 1, sample space Ω, events as subsets, complement P(A') = 1 − P(A), union A ∪ B, intersection A ∩ B, exhaustive events, sum rule, theoretical vs empirical probability, frequentist vs Bayesian vs subjective interpretations, principle of indifference.
• The rules — addition rule for mutually exclusive events, multiplication rule for independent and for dependent events, conditional probability, general multiplication rule, Bayes theorem, law of total probability, joint and marginal probability, prior, posterior, likelihood function.
• Random variables — discrete vs continuous, PMF, PDF, CDF F(x) = P(X ≤ x), f(x) = dF/dx, expected value as Σ x·P(x) and as ∫ x·f(x) dx, variance E(X²) − [E(X)]², σ = √Var(X), Σ P(x) = 1, ∫ f(x) dx = 1.
• Distributions — Bernoulli trials, binomial (μ = n·p, σ² = n·p·(1−p)), Poisson with rate λ, the normal PDF, standard normal N(0,1), z-scores, the 68-95-99.7 empirical rule, uniform on [a, b], exponential waiting times, memorylessness.
• Counting and limit theorems — permutations P(n,r) = n!/(n−r)!, combinations C(n,r), multiplication principle, sampling with and without replacement, law of large numbers, central limit theorem, Markov chains, tree diagrams, contingency tables.
• Paradoxes — gambler's fallacy, birthday problem, Monty Hall, Simpson's paradox, St. Petersburg paradox, fair games with expected value zero, Laplace's demon.

Two tips for this deck

1. Card 60 starts a second deck — prune it. The first 60 cards are an ordered curriculum; the rest re-teaches about a dozen ideas. The complement rule appears twice, so do the general addition rule, expected value, the normal distribution, Poisson, exponential and Bernoulli; the sum rule returns as Σ P(x) = 1; the empirical rule is taught as three ±1σ/±2σ/±3σ cards, then again as one. Suspend one of each pair and the honest total drops to roughly 88. Keep the rest of the tail: Monty Hall, the birthday problem, Markov chains, Simpson's paradox, odds vs probability and the prior/posterior/likelihood trio appear nowhere else.

2. Rewrite the central limit theorem card. Its back says sample means approach normality "regardless of the population distribution". Two conditions are missing: finite variance — the theorem fails outright for Cauchy-type heavy tails — and a large enough n, usually n ≥ 30 and more when the population is strongly skewed. Add both, because exam questions target exactly that gap.

Where to find it

Log in at flip-it.app and search the library for code EX3246. Pair it with the descriptive half of the same syllabus: How to Master Descriptive Statistics.