Probability: dice, cards, Bayes

Dice — standard notation: 3d6, 1d20+5, 2d6-1d4, 4d6dl1 (drop lowest), 2d20kh1 (advantage), 2d20kl1 (disadvantage), d%, and Fudge dice 4dF. The distribution is exact: each term’s outcomes are convolved, and keep/drop dice are tracked through their kept values. Set a target to see P(total ≥ target).

Combinations — nCr, nPr, with repetition (nʳ), and multisets, computed exactly with arbitrary-size integers.

Cards and lottery — five-card poker hand frequencies out of C(52,5) = 2 598 960; a lottery’s odds for each number of matches, C(k,m)·C(n−k,k−m)/C(n,k); and the hypergeometric distribution for draws without replacement.

Birthday problem — P(shared) = 1 − Π(d − i)/d. With 365 days, 23 people give 50.7 %. Change the days to estimate hash or ID collisions.

Distributions — binomial, geometric (trials until the first success), and Poisson, with exact and cumulative probabilities.

Bayes — P(H|) = P(|H)·P(H)/P(+), shown as a tree of whole people and an icon array of 1,000 people (true and false positives, misses, true negatives). That makes it clear why a good test for a rare condition still gives mostly false positives. Slide the prevalence from 50 % down to 0.1 % on a log scale, and switch to two independent positives to see the first posterior become the second prior.

Bloom filter — sliders for the bits m, items n and hash functions k. It shows the fill 1 − e^(−kn/m), the false-positive rate (1 − e^(−kn/m))^k (and the exact form without the e approximation), the optimal k = (m/n) ln 2 and the best whole k, a log-scale curve of the false-positive rate against k, and a simulation: insert n items, query 20,000 that were never added, and compare measured with predicted. Filters up to 4,096 bits draw their bit array.

Spam filter — a naive Bayes classifier you train on a few spam and ham messages (one per line). Each word of the message gets P(word | spam) and P(word | ham) as the fraction of that class’s messages containing it, with add-α smoothing (hits + α)/(messages + 2α). The table shows every factor and its log ratio; the scores, their log difference and P(spam | message) follow. With α = 0, one word never seen in a class zeroes that class and the tool says so.

Reliability — describe a system of independent parts: s(…) in series (all must work), p(…) in parallel (any one is enough), kofn(2, …) voting, 0.99x3 for three identical parts, 99% for 0.99; nest freely, e.g. s(0.999, p(0.99x2), p(0.95x3)). You get the reliability, failure probability, "nines" and downtime per year, a block diagram, and each group’s calculation from the inside out. Below it, failures at scale: N units with a given MTBF fail together every MTBF ÷ N, with the failures per day, the expected number in a run, P(no failure) = e^(−N·T/MTBF), and how much time goes to recovery under a hot standby (tens of ms, twice the GPUs), warm standby (about a second, a shared cache tier) or cold restart (re-run the prefill).

Expected value and Monte Carlo — list outcomes as "probability, value". For anything harder, write one trial as JavaScript; it runs millions of times in a background Worker (stopped after 20 s) and reports the hit rate with a 95 % interval.