About

Probability is the mathematical framework for quantifying uncertainty. While basic calculators handle coin flips, professional analysis requires understanding complex distributions, conditional events, and combinatorial logic. This tool serves as a comprehensive engine for students, data analysts, and risk managers. It computes theoretical probabilities for six distinct statistical models, including Binomial Distribution and Bayesian Inference, providing exact decimal, fractional, and percentage outputs.

Theoretical precision is often disconnected from observed reality. To bridge this gap, the integrated Monte Carlo Simulator executes up to 10,000 distinct trials for your specific scenario. This experimental approach demonstrates the Law of Large Numbers, visually plotting how short-term variance eventually converges to the mathematical expectation. Whether calculating poker odds, server failure rates, or A/B testing significance, this tool provides both the formulaic proof and the experimental validation.

Formulas

Advanced probability relies on combinatorial logic and set theory. Below are the core models used in this engine.

1. Combinations (Order doesn't matter): Used for lottery odds and card hands.

C(n,k) = n!k!(n − k)!

2. Permutations (Order matters): Used for passwords, races, and ranking.

P(n,k) = n!(n − k)!

3. Binomial Distribution: Calculates exactly k successes in n independent trials with probability p.

P(X=k) = nk p^k (1 − p)^n−k

4. Bayes' Theorem: Updates the probability of an event based on prior knowledge.

P(A|B) = P(B|A)P(A)P(B)

Reference Data

Event Category	Specific Scenario	Odds (Ratio)	Probability (Dec)	Probability (%)
Gambling (Cards)	Royal Flush (5-Card Stud)	1 : 649,739	0.00000154	0.000154%
Gambling (Cards)	Four of a Kind	1 : 4,164	0.000240	0.024%
Gambling (Cards)	Full House	1 : 693	0.00144	0.144%
Gambling (Dice)	Snake Eyes (1 and 1 on 2d6)	1 : 35	0.0278	2.78%
Gambling (Roulette)	Single Number (European)	1 : 36	0.0270	2.70%
Gambling (Roulette)	Red or Black (American)	1 : 1.11	0.4737	47.37%
Lottery	Powerball Jackpot	1 : 292,201,338	3.42 × 10^-9	0.0000003%
Lottery	Mega Millions Jackpot	1 : 302,575,350	3.30 × 10^-9	0.0000003%
Nature	Struck by Lightning (Lifetime)	1 : 15,300	0.000065	0.0065%
Nature	Born on Leap Day (Feb 29)	1 : 1,460	0.00068	0.068%
Nature	Giving Birth to Identical Twins	1 : 250	0.004	0.4%
Health & Safety	Left-Handedness (Global)	1 : 9	0.10	10.0%
Health & Safety	O-Negative Blood Type	1 : 14	0.066	6.6%
Health & Safety	Car Accident (Annual/Driver)	1 : 366	0.0027	0.27%
Technology	Hard Drive Failure (Annual)	1 : 50	0.02	2.0%
Technology	Email Opening Rate (Avg)	1 : 4	0.213	21.3%
Technology	Ecommerce Conversion Rate	1 : 33	0.03	3.0%
Space	Meteorite Impact (Person/Year)	1 : 1,600,000	6.25 × 10^-7	0.00006%
Sports	Hole in One (Amateur Golfer)	1 : 12,500	0.00008	0.008%
Sports	Perfect Game (MLB History)	1 : 19,000	0.00005	0.005%

Frequently Asked Questions

The key difference is order. In Permutations (nPr), the order of selection matters (e.g., a combination lock code 1-2-3 is different from 3-2-1). In Combinations (nCr), order does not matter (e.g., a lottery ticket 1-2-3 is the same winning ticket as 3-2-1).

Use Binomial Distribution when you have a fixed number of independent trials (n), each trial has only two outcomes (success/failure), and the probability of success (p) remains constant. Examples include coin flips, manufacturing defect rates, or hitting a target multiple times.

The simulator runs 10,000 iterations. While theoretical math gives the exact "infinite" average, the simulator shows what might happen in the real world. As the number of runs increases, the experimental result converges toward the theoretical probability, a concept known as the Law of Large Numbers.

Probability is a fraction of the total (Success / Total). Odds compare Successes directly to Failures (Success : Failure). For example, a 20% probability (1/5) translates to 1:4 odds (1 win for every 4 losses).

Bayes' Theorem is used to update the probability of a hypothesis as more evidence becomes available. It is widely used in medical diagnosis (probability of disease given a positive test) and spam filtering (probability of spam given certain words).