Probability Calculator

**Basic probability:** P(event) = Favorable outcomes / Total outcomes For equally likely outcomes only. **Worked example: dice roll** P(rolling a 4) = 1/6 ≈ 0.167 = 16.7% Odds: 1:5 in favor. **Properties of probability:** - **Range**: 0 ≤ P ≤ 1 - **Sum of all outcomes**: ΣP(xᵢ) = 1 - **Complement**: P(not A) = 1 - P(A) - **Impossible event**: P = 0 - **Certain event**: P = 1 **Probability rules:** **Independent events** (one doesn't affect the other): P(A and B) = P(A) × P(B) Example: rolling 4 twice in a row: 1/6 × 1/6 = 1/36. **Dependent events**: P(A and B) = P(A) × P(B|A) Where P(B|A) is conditional probability of B given A. **Mutually exclusive events** (can't both happen): P(A or B) = P(A) + P(B) Example: rolling 1 or 6 on a die: 1/6 + 1/6 = 2/6 = 1/3. **Non-mutually exclusive events**: P(A or B) = P(A) + P(B) - P(A and B) **Conditional probability:** P(A|B) = P(A and B) / P(B) "Probability of A given B." **Bayes' theorem:** P(A|B) = P(B|A) × P(A) / P(B) Useful for updating beliefs when new information arrives. **Probability vs odds:** | Probability | Odds in favor | Odds against | |---|---|---| | 0.10 | 1:9 | 9:1 | | 0.25 | 1:3 | 3:1 | | 0.50 | 1:1 | 1:1 | | 0.75 | 3:1 | 1:3 | | 0.90 | 9:1 | 1:9 | **Converting between:** - Probability → Odds in favor: P / (1-P) - Probability → Odds against: (1-P) / P - Odds in favor (a:b) → Probability: a / (a+b) - Odds against (a:b) → Probability: b / (a+b) **Common probability scenarios:** **Coin flip:** - P(heads) = 1/2 = 0.5 - P(2 heads in 2 flips) = 0.5 × 0.5 = 0.25 - P(at least 1 head in 2 flips) = 1 - P(no heads) = 1 - 0.25 = 0.75 **Card deck (52 cards):** - P(any specific card) = 1/52 ≈ 0.019 - P(face card) = 12/52 = 3/13 ≈ 0.231 - P(red card) = 26/52 = 1/2 = 0.5 - P(heart) = 13/52 = 1/4 = 0.25 **Dice:** - P(sum of 2 dice = 7) = 6/36 = 1/6 - P(sum = 12) = 1/36 - P(at least one 6 in 2 dice) = 11/36 **Lottery probabilities:** - 6 from 49: 1 / C(49,6) = 1/13,983,816 ≈ 7.15 × 10⁻⁸ - Powerball: 1 / 292,201,338 - Mega Millions: 1 / 302,575,350 **Common applications:** - **Gambling**: house edge, game odds. - **Insurance**: actuarial calculations. - **Sports**: betting odds, game outcome predictions. - **Medicine**: drug effectiveness, disease prevalence. - **Finance**: risk assessment, options pricing. - **Machine learning**: Bayesian inference, classification. - **Quality control**: defect rates. **Probability fallacies:** 1. **Gambler's fallacy**: thinking past results affect future random events. P(heads | 10 tails in row) = 0.5, not higher. 2. **Hot hand fallacy**: similar — believing in streaks where there are none. 3. **Conjunction fallacy**: thinking P(A and B) > P(A). False — P(A and B) ≤ P(A) always. 4. **Base rate neglect**: ignoring the underlying rate when estimating probabilities. Common in medical decision-making. 5. **Confirmation bias**: noticing evidence that supports beliefs while ignoring contradicting evidence. **Independent vs dependent:** - **Independent**: P(A and B) = P(A) × P(B). Events don't affect each other. - **Dependent**: P(A and B) = P(A) × P(B|A). One affects the other. Example: drawing cards without replacement is dependent; coin flips are independent. **Probability distributions:** | Distribution | Use | |---|---| | Bernoulli | Single trial (success/failure) | | Binomial | n independent trials | | Geometric | Number of trials until first success | | Poisson | Rare events in fixed period | | Negative binomial | Trials until r successes | | Hypergeometric | Sampling without replacement | | Multinomial | n trials, k outcomes | | Normal (continuous) | Many natural variables | | Exponential | Time between events | | Beta | Probability of probability | **Subjective vs objective probability:** - **Objective**: based on physical mechanism (fair dice, well-shuffled cards) or empirical data. - **Subjective**: based on belief and information available. Bayesian approach embraces this. - Both are valid; choice depends on application.