What is Probability?

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain), or as percentage 0%-100%. Foundation of statistics, gambling, risk assessment, machine learning. Five modes supported: Basic (favorable / total), Both A AND B (joint probability of two independent events), Either A OR B (union of two events), Conditional P(A given B happened), Complement (1 minus probability). Used for: assessing risks (insurance, investment), gambling odds (dice, cards), medical diagnosis (positive/negative predictive value), weather forecasting, A/B test analysis, decision theory.

How to use this tool

1. Choose probability type — Basic, Both, Either, Conditional, or Complement.
2. Enter values — Either counts (favorable/total) or probabilities (0-1).
3. Calculate — Result shown as decimal and percentage.
4. Interpret — 0 = impossible, 0.5 = 50% chance, 1 = certain.

Probability formulas

• Basic: P(event) = favorable outcomes / total outcomes
• Both independent: P(A and B) = P(A) × P(B)
• Either independent: P(A or B) = P(A) + P(B) – P(A and B)
• Conditional: P(A | B) = P(A and B) / P(B) — probability of A given B happened
• Complement: P(not A) = 1 – P(A)

Bayes' Theorem (related):

P(A | B) = [P(B | A) × P(A)] / P(B)

Used for diagnostic tests: P(disease | positive test).

Examples

• Coin flip heads: P = 1/2 = 0.5 = 50%
• Two coin flips both heads: P = 0.5 × 0.5 = 0.25 = 25%
• Dice roll 6: P = 1/6 ≈ 16.67%
• Two dice both 6: P = 1/36 ≈ 2.78%
• Draw queen of spades: P = 1/52 ≈ 1.92%
• Either red OR ace from deck: 26/52 + 4/52 – 2/52 = 28/52 ≈ 53.85%

Tips & best practices

• Use decimal 0-1 for calculations; convert to percentage for communication
• AND multiplies (events happen together); OR adds (at least one happens)
• Independent events: outcome of one doesn't affect other (coin flips, separate dice)
• Dependent events: outcome affects next (cards drawn without replacement)
• Conditional probability is foundation of Bayesian statistics, AI/ML, medical diagnostics

Frequently Asked Questions

What if probabilities sum > 1?

Invalid — means events overlap (not mutually exclusive). For OR: subtract overlap. For Both: use multiplication for independents.

When is conditional probability useful?

Medical tests: P(disease | positive test) tells you actual disease risk given test. P(test positive | disease) is sensitivity. Different things — conditional probability connects them.

Coin flips are independent?

Yes — each flip's outcome doesn't affect next. P(heads twice) = P(heads) × P(heads) = 0.25, NOT 0.5.

What is 'gambler's fallacy'?

Belief that past outcomes affect future independent events. 'Roulette landed red 5 times, next must be black!' — FALSE. Each spin independent with 18/38 red probability.

How to calculate lottery odds?

Total combinations / favorable combinations. Indian Lotto 6/49: 13,983,816 total combinations — your one ticket has 1/13,983,816 chance.

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