Bayes’ Theorem and Base Rates

/**
 * Fintech Math Bootcamp · Lesson 055 of 120
 * Bayes’ Theorem and Base Rates
 * Module 06: Probability and Random Variables
 *
 * Scenario: Why a good fraud detector can still generate many false alarms
 * Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
 *
 * Try it:   How many false flags arise in the 10,000-transaction example?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson055() {
  const prior=.01, sensitivity=.90, falsePositiveRate=.05;
  const truePositiveMass=sensitivity*prior;
  const falsePositiveMass=falsePositiveRate*(1-prior);
  const result = truePositiveMass/(truePositiveMass+falsePositiveMass);
  return result;
}

export const checkedResult = 0.15384615384615385;

// Run this file directly: npx tsx lessons/06-probability-and-random-variables/055-bayes-theorem-and-base-rates.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson055(), null, 2));
}

# Fintech Math Bootcamp · Lesson 055 of 120
# Bayes’ Theorem and Base Rates
# Module 06: Probability and Random Variables
#
# Scenario: Why a good fraud detector can still generate many false alarms
# Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
#
# Try it:   How many false flags arise in the 10,000-transaction example?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.

import json


def lesson055() -> float:
    prior, sensitivity, false_positive_rate = 0.01, 0.90, 0.05
    true_positive_mass = sensitivity * prior
    false_positive_mass = false_positive_rate * (1 - prior)
    return true_positive_mass / (true_positive_mass + false_positive_mass)


if __name__ == "__main__":
    print(json.dumps(lesson055(), indent=2))

/**
 * Fintech Math Bootcamp · Lesson 055 of 120
 * Bayes’ Theorem and Base Rates
 * Module 06: Probability and Random Variables
 *
 * Scenario: Why a good fraud detector can still generate many false alarms
 * Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
 *
 * Try it:   How many false flags arise in the 10,000-transaction example?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

import java.util.ArrayList;
import java.util.List;
import java.util.Map;

public class Main {
    // P(fraud | flag) from the base rate, sensitivity and false-positive rate.
    static double lesson055() {
        double prior = 0.01, sensitivity = 0.90, falsePositiveRate = 0.05;
        double truePositiveMass = sensitivity * prior;
        double falsePositiveMass = falsePositiveRate * (1 - prior);
        return truePositiveMass / (truePositiveMass + falsePositiveMass);
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson055(), ""));
    }

    // --- Minimal JSON printer: maps keep insertion order, 2-space indent. ---

    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return "\"" + value + "\"";
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        if (value instanceof double[]) {
            List<Object> items = new ArrayList<Object>();
            for (double v : (double[]) value) items.add(v);
            return toJson(items, indent);
        }
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                sb.append(inner).append('"').append(entry.getKey()).append("\": ")
                  .append(toJson(entry.getValue(), inner))
                  .append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append('}').toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner))
              .append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append(']').toString();
    }

    static String formatNumber(double v) {
        if (Double.isNaN(v) || Double.isInfinite(v)) return "null";
        if (v == Math.rint(v) && Math.abs(v) < 1e15) return Long.toString((long) v);
        return Double.toString(v);
    }
}

// Fintech Math Bootcamp · Lesson 055 of 120
// Bayes’ Theorem and Base Rates
// Module 06: Probability and Random Variables
//
// Scenario: Why a good fraud detector can still generate many false alarms
// Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
//
// Try it:   How many false flags arise in the 10,000-transaction example?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

package main

import (
	"encoding/json"
	"fmt"
)

// lesson055 returns P(fraud | flag) from the base rate, sensitivity and false-positive rate.
func lesson055() float64 {
	prior, sensitivity, falsePositiveRate := 0.01, 0.90, 0.05
	truePositiveMass := sensitivity * prior
	falsePositiveMass := falsePositiveRate * (1 - prior)
	return truePositiveMass / (truePositiveMass + falsePositiveMass)
}

func main() {
	out, _ := json.MarshalIndent(lesson055(), "", "  ")
	fmt.Println(string(out))
}

/**
 * Fintech Math Bootcamp · Lesson 055 of 120
 * Bayes’ Theorem and Base Rates
 * Module 06: Probability and Random Variables
 *
 * Scenario: Why a good fraud detector can still generate many false alarms
 * Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
 *
 * Try it:   How many false flags arise in the 10,000-transaction example?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

#include <charconv>
#include <cmath>
#include <iostream>
#include <string>
#include <utility>
#include <vector>

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

struct Json {
    enum class Kind { Null, Number, Text, Array, Object };
    Kind kind = Kind::Null;
    double number = 0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values
};

Json num(double v) { Json j; j.kind = Json::Kind::Number; j.number = v; return j; }
Json str(const std::string& s) { Json j; j.kind = Json::Kind::Text; j.text = s; return j; }
Json arr(const std::vector<Json>& values) { Json j; j.kind = Json::Kind::Array; j.items = values; return j; }

Json arr(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(num(v));
    return arr(items);
}

Json obj(const std::vector<std::pair<std::string, Json>>& fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (const auto& [key, value] : fields) { j.keys.push_back(key); j.items.push_back(value); }
    return j;
}

std::string formatNumber(double v) {
    if (!std::isfinite(v)) return "null";
    char buf[64];
    auto end = std::to_chars(buf, buf + sizeof buf, v).ptr;  // shortest round-trip form
    return std::string(buf, end);
}

void writeJson(std::ostream& out, const Json& j, const std::string& indent) {
    switch (j.kind) {
    case Json::Kind::Null: out << "null"; return;
    case Json::Kind::Number: out << formatNumber(j.number); return;
    case Json::Kind::Text: out << '"' << j.text << '"'; return;
    default: break;
    }
    bool isObject = j.kind == Json::Kind::Object;
    if (j.items.empty()) { out << (isObject ? "{}" : "[]"); return; }
    std::string inner = indent + "  ";
    out << (isObject ? "{\n" : "[\n");
    for (size_t i = 0; i < j.items.size(); ++i) {
        out << inner;
        if (isObject) out << '"' << j.keys[i] << "\": ";
        writeJson(out, j.items[i], inner);
        out << (i + 1 < j.items.size() ? ",\n" : "\n");
    }
    out << indent << (isObject ? '}' : ']');
}

// --- Lesson ---

// P(fraud | flag) from the base rate, sensitivity and false-positive rate.
Json lesson055() {
    const double prior = 0.01, sensitivity = 0.90, falsePositiveRate = 0.05;
    const double truePositiveMass = sensitivity * prior;
    const double falsePositiveMass = falsePositiveRate * (1 - prior);
    return num(truePositiveMass / (truePositiveMass + falsePositiveMass));
}

int main() {
    writeJson(std::cout, lesson055(), "");
    std::cout << '\n';
}

// Fintech Math Bootcamp · Lesson 055 of 120
// Bayes’ Theorem and Base Rates
// Module 06: Probability and Random Variables
//
// Scenario: Why a good fraud detector can still generate many false alarms
// Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
//
// Try it:   How many false flags arise in the 10,000-transaction example?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

#[allow(dead_code)]
enum Json {
    Null,
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

#[allow(dead_code)]
fn nums(values: &[f64]) -> Json {
    Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
}

#[allow(dead_code)]
fn obj(fields: Vec<(&str, Json)>) -> Json {
    Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
}

fn format_number(v: f64) -> String {
    if !v.is_finite() {
        return "null".to_string();
    }
    if v.fract() == 0.0 && v.abs() < 1e15 {
        return format!("{}", v as i64);
    }
    format!("{:?}", v) // shortest round-trip form
}

impl Json {
    fn render(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Num(v) => format_number(*v),
            Json::Str(s) => format!("\"{}\"", s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let lines: Vec<String> = items.iter().map(|v| format!("{}{}", inner, v.render(&inner))).collect();
                format!("[\n{}\n{}]", lines.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let lines: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}\"{}\": {}", inner, k, v.render(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", lines.join(",\n"), indent)
            }
        }
    }
}

// --- Lesson ---

/// P(fraud | flag) from the base rate, sensitivity and false-positive rate.
fn lesson055() -> Json {
    let (prior, sensitivity, false_positive_rate): (f64, f64, f64) = (0.01, 0.90, 0.05);
    let true_positive_mass = sensitivity * prior;
    let false_positive_mass = false_positive_rate * (1.0 - prior);
    Json::Num(true_positive_mass / (true_positive_mass + false_positive_mass))
}

fn main() {
    println!("{}", lesson055().render(""));
}

/**
 * Fintech Math Bootcamp · Lesson 055 of 120
 * Bayes’ Theorem and Base Rates
 * Module 06: Probability and Random Variables
 *
 * Scenario: Why a good fraud detector can still generate many false alarms
 * Rule:     posterior = sensitivity·prior / [sensitivity·prior + FPR·(1−prior)]
 *
 * Try it:   How many false flags arise in the 10,000-transaction example?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/bayes-theorem-and-base-rates/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

using System;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson055(), options));

// P(fraud | flag) from the base rate, sensitivity and false-positive rate.
static double Lesson055()
{
    double prior = 0.01, sensitivity = 0.90, falsePositiveRate = 0.05;
    double truePositiveMass = sensitivity * prior;
    double falsePositiveMass = falsePositiveRate * (1 - prior);
    return truePositiveMass / (truePositiveMass + falsePositiveMass);
}