Math, Applied

Exploring how math and data shape real-world decisions.

Breaking down complex systems into ideas we can actually use.

Ganita

Hi, I'm Ganita. Describe a decision or data problem in plain language and I'll point you to the best posts here.

Example: “We ran twenty A/B tests and three won. Should we ship them?”

Math foundations

Core probability, distributions, EV, and lines that unlock the business posts.

Probability Basics: Events, Joint Probability, and Independence

Pure probability building blocks before win rates, base rates, and screening.

Conditional Probability: Given That B Happened

P(A | B) versus P(B | A): restrict the sample space before you read a screen or alert.

Dependent Chains: Funnel Math Without Bogus Independence

P(step1) × P(step2 | step1) for funnels, pipelines, and onboarding paths.

Sensitivity, Specificity, and the 2×2 Screening Table

TP, FP, FN, TN and positive predictive value before base rates and thresholds.

Reading Distributions: Percentiles and Quartiles from Scratch

Sort data, read cutoffs, and understand P90 and quartiles before SLA metrics.

Expected Value: The Pure Definition

EV as probability-weighted average before campaign and pipeline decisions.

Linear Models: y = a + bx in Plain Math

Intercept, slope, and residuals before regression forecasting posts.

Vectors and Features: One Row Is a Point in Space

Spreadsheet rows as feature vectors: dimensions, coordinates, and the design matrix behind ML.

Dot Product and Cosine Similarity: How Aligned Are Two Vectors?

Dot product vs cosine for search, recommendations, and RAG when vector length varies.

Matrices and Linear Systems: Regression Is Ax = b

Normal equations and design matrices: the linear system regression actually solves.

Least Squares: Why Squared Error Picks One Best Line

Sum of squared residuals and why the best-fit line is the unique SSE minimizer.

Norms and Distance: How Far Apart Are Two Feature Vectors?

L1 and L2 distance for anomaly detection, duplicate search, and comparing rows in feature space.

Orthogonality: When Two Features Carry Separate Signal

Correlation as angle: orthogonal inputs read cleanly in regression; parallel inputs share credit.

k-Nearest Neighbors: Classify by the Closest Labeled Examples

Distance-based voting: fraud similarity, routing, and duplicate detection without a training phase.

Logistic Regression: Linear Boundary, Probability Score

Sigmoid scores, linear decision boundaries, and the bridge from regression to classification.

Classification Loss: What Optimizers Actually Minimize

Cross-entropy, hinge, and 0-1 loss: why training uses smooth penalties, not accuracy.

Gradient Descent: How Classifiers Learn Their Weights

Learning rate, gradient steps, and SGD intuition for fitting logistic and neural models.

Permutations: Order Matters When You Count Arrangements

P(n,r) counting when sequence matters: rankings, seatings, and ordered pipelines.

Combinations: Choosing a Set When Order Does Not Matter

n choose k for subsets, feature picks, and committees when order does not change the outcome.

Determinants: Area Scale Factor of a Linear Map

det(A) as signed area scale, singularity when det is zero, and invertibility of linear systems.

Eigenvalues: Stretch Factors Along Special Directions

λ in Av = λv: stretch factors, characteristic equation, and PCA variance along components.

Eigenvectors: Directions That Only Stretch, Not Rotate

Directions unchanged in angle under a linear map: principal axes and independent variance.

Feature Scaling: Why Dollars Crush Session Counts Without Z-Scores

Z-scores and min-max scaling so distance, k-NN, and gradients are not dominated by unit choice.

Bayes Theorem: Update Belief When New Evidence Arrives

P(A|B) from prior and likelihood: screening, alerts, and flipping evidence into posteriors.

Confusion Matrix: The Four Boxes Behind Every Classifier Score

TP, FP, FN, TN and how precision, recall, and accuracy are built from the 2x2 table.

The Normal Distribution: Bell Curves, Mean, and Standard Deviations

Gaussian density, the 68-95-99.7 rule, and when a bell curve is a useful approximation.

Logarithms and Odds: Why Log-Odds Show Up in Logistic Models

Odds, logit, and log scales that turn multiplication into addition in logistic regression.

Covariance: How Two Features Move Together

Cov(X,Y), its units, and the bridge to correlation and the covariance matrix.

Summarize the data

Means, spread, percentiles, and when one number misleads.

Prime Factorization Isn't Just Math: It's How You Break Down Real Problems

Clean batch splits, ETL jobs, and resource planning from number structure.

The Average Isn't the Answer: Mean and Median in Real Data

When one summary number misleads dashboards and product decisions.

Percent Change Isn't Intuitive: Growth Math in Real Decisions

Why recovery percentages do not undo drops, and how to set targets.

Variance and Spread: Same Average, Different Story

Standard deviation and range for delivery, ops, and reliability.

PCA Intuition: One Axis for a Wall of Correlated KPIs

Principal components as composite scores when dashboard metrics move together.

The Average User Isn't Average: Percentiles and Quartiles

P90, SLAs, latency tails, and typical vs worst-case experience.

Weighted Averages: Roll Up the Number That Matches Volume

Simple vs volume-weighted rollups when segment sizes differ.

Up 10% Compared to What? Benchmarks and Baselines

Same metric, different story: last month, last year, plan, or peer.

The New Users Look Great: Cohort Analysis in Real Decisions

Headline retention vs signup cohorts: when the mix shifts and when product quality actually changed.

More Tickets, Same Workload? Rates vs Counts in Real Data

Total volume vs per-user rate: when headline counts rise because the base grew.

Up From Last Month: Is That Normal for March? Seasonality

Month-over-month vs same month last year when the calendar drives the metric.

Experiments and uncertainty

Sample size, probability, intervals, and when to ship.

Bias vs Variance: Underfit, Overfit, and the U-Shaped Error Curve

Expected error as bias squared plus variance: complexity sweet spots before you ship.

Twelve Data Points Isn't a Trend: Sample Size in Real Decisions

A/B tests, noisy averages, and when a readout is too thin to ship.

We Ran the Test: Could We Even See a Win? Statistical Power

Detectability before you launch: sample size, baseline, and minimum lift together.

Probability in Real Decisions: Count Wins, Not Stories

Estimate win rates from trials and know when the read is still shaky.

At Least One Failure: When Small Risks Compound

P(at least one) = 1 − (1 − p)^n for deploys, SLA misses, and repeated risky trials.

Stacking Rare Risks: Alert Fatigue From Many Small False Positives

Many low false-positive checks on one case: system alert rate is 1 − (1 − f)^k.

Will We Run Out? Probability of Stockout in Real Inventory Decisions

Set reorder points using explicit stockout risk instead of gut feel.

One Number Is Not Enough: Confidence Intervals

Ranges around conversion, CSAT, and defect rates before you ship.

Base Rates and Updating Beliefs: Rare Events, Loud Alerts

Why a strong alert can still mean mostly false alarms when the base rate is low.

Classifier Metrics: Precision, Recall, Accuracy, and AUC in Plain Language

TP, FP, FN, TN, every major classifier score, precision vs recall tradeoffs, and draggable ROC/AUC charts.

Decision Trees: Readable Rules for Classification

Axis-aligned splits, readable if-then policies, and depth tradeoffs for fraud and hiring screens.

The Model Says 90%: Can You Trust the Score?

Reliability diagrams, overconfident scores, and why accuracy can look fine while auto-block rules fail.

Where Do You Draw the Line? Threshold Tradeoffs in Real Decisions

Precision vs recall, review capacity, and dollar cost when you pick a classifier cutoff.

How Many Labels Before You Trust the Metric?

Labeled-set size for precision, recall, and F1 — when the eval band is too wide to ship.

A/B Test Readouts: Significance Without Jargon

Lift, sample size, interval overlap, and when to ship without p-value talk.

When Everything Wins Once: Running Many A/B Tests

False winners multiply when you run dozens of null tests in one sprint.

Simpson's Paradox: When Every Slice Wins but the Total Loses

Segment tables before rollups so mix shifts do not flip the winner.

False Alarm vs Missed Win: Two Ways an Experiment Decision Goes Wrong

Ship a bad change vs kill a good one: pick which mistake you can afford.

Expected Value: Compare Bets Without Guessing

Rank campaigns by upside, probability, and cost before spend.

Traps and causation

Bias, confounders, and conclusions that look right but are not.

Last Quarter's Model, This Quarter's Data: Concept Drift

When live behavior shifts but training accuracy still looks fine — and auto-decisions start misfiring.

Correlation Isn't Causation: Linked Data in Real Decisions

Confounders, direction, and what you need before acting on r.

Multicollinearity: When Two Drivers Share the Same Story

Correlated regression inputs make coefficients unstable and driver credit unreliable.

Ridge Regularization: Shrink Unstable Coefficients Without Dropping Features

Ridge penalty stabilizes collinear regression inputs by shrinking coefficients toward zero.

The Gambler's Fallacy: Streaks Do Not Load the Next Trial

Five losses in a row does not make the next independent trial more likely to win.

Your Best Customers Answered: Selection Bias

Surveys, betas, and who never made it into the sample.

Only the Winners Stay Visible: Survivorship Bias

Success stories hide failures that left the dataset before you measured.

You Hit the Target and Missed the Point: Goodhart's Law

When the metric becomes the goal, the dashboard greens while the real outcome reddens.

The Star Performer Slump: Regression to the Mean

Sales quotas, support metrics, and snap-back after extreme scores.

The Model Looked Perfect on Past Data: Overfitting in Real Decisions

Train vs holdout: when forecasts memorize history instead of predicting new weeks.

Regression for Prediction: Data to Decisions

Weekly orders from ad spend and email, with forecast, holdout, and scenarios.

Case studies

Long decision stories with tradeoffs, explorers, and links to the underlying math.

Company X: The Weekend Reorder Decision

A full case walkthrough at Company X: festival demand, three teams, reorder points, and explicit stockout risk.