Probability Calculator

Probability measures how likely an event is to occur. It is always between 0 (impossible) and 1 (certain). Formula: P(event) = number of favorable outcomes / total possible outcomes. Example: probability of rolling a 4 on a standard die = 1/6 ≈ 0.1667 ≈ 16.67%. Multiply by 100 to express as a percentage. The sum of probabilities of all possible outcomes in a sample space always equals exactly 1.

The complement of event A is "A does not occur," written as A' or Aᶜ. P(A') = 1 − P(A). If the probability of rain tomorrow is 0.3 (30%), then the probability of no rain = 1 − 0.3 = 0.7 (70%). Complements are useful because sometimes it is easier to calculate the probability of an event NOT occurring and subtract from 1. Example: probability of getting at least one head in 3 flips = 1 − P(no heads) = 1 − (0.5)³ = 1 − 0.125 = 0.875.

AND probability (intersection): both events occur. For independent events: P(A and B) = P(A) × P(B). Example: flipping heads AND rolling a 6 = 0.5 × 1/6 ≈ 0.083. OR probability (union): at least one event occurs. P(A or B) = P(A) + P(B) − P(A and B). Example: drawing a heart OR a face card from a deck = 13/52 + 12/52 − 3/52 = 22/52 ≈ 42.3%. The subtraction avoids double-counting outcomes where both occur.

Independent events: the outcome of one does not affect the other. Example: flipping a coin and rolling a die are independent. P(A and B) = P(A) × P(B). Mutually exclusive events: both cannot occur at the same time. Example: rolling a 1 and rolling a 2 on one die are mutually exclusive. P(A and B) = 0, so P(A or B) = P(A) + P(B). Caution: mutually exclusive events are NOT independent (if A occurs, B cannot, so they are dependent on each other).

Binomial probability calculates the chance of getting exactly k successes in n independent trials, where each trial has probability p of success. Formula: P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k). Example: probability of getting exactly 3 heads in 10 coin flips. P(X=3) = C(10,3) × 0.5³ × 0.5&sup7; = 120 × 0.125 × 0.0078125 ≈ 0.117 = 11.7%. The binomial distribution describes many real-world scenarios: defect rates, survey responses, medical test results.

Permutations (nPr): arrangements where order matters. nPr = n!/(n−r)!. Example: how many 3-digit PINs from digits 1–9 (no repeats)? 9P3 = 9!/6! = 504. Combinations (nCr): selections where order does not matter. nCr = n!/(r!(n−r)!). Example: choosing 3 people from 9 for a committee (the order of selection doesn't matter): C(9,3) = 84. Relationship: nPr = nCr × r! (permutations = combinations × the number of ways to arrange the chosen items).

Odds express the ratio of favorable to unfavorable outcomes. Odds in favor of A = P(A) / P(A') = favorable / unfavorable. Example: rolling a 6 has probability 1/6. Odds in favor = 1:5 (1 favorable, 5 unfavorable). Odds against = 5:1. Converting: if odds in favor = a:b, then P = a/(a+b). If odds = 3:1 in favor, P = 3/4 = 0.75. Probability and odds describe the same information differently. Odds are commonly used in gambling and sports betting, while probability is used in science and statistics.

Conditional probability P(A|B) is the probability that A occurs given that B has already occurred. Formula: P(A|B) = P(A and B) / P(B). Example: a bag has 3 red and 2 blue balls. Probability of drawing red on the second draw given red on the first (without replacement): P(red first) = 3/5. P(red first AND red second) = 3/5 × 2/4 = 6/20 = 3/10. P(red second | red first) = (3/10) / (3/5) = 1/2. Bayes' theorem uses conditional probability to update beliefs based on new evidence.

Insurance: companies calculate the probability of claims to set premiums. Medicine: clinical trials measure the probability that a treatment works vs placebo. Weather forecasting: "70% chance of rain" is a probability statement. Finance: options pricing (Black-Scholes) uses probability that an asset reaches a certain price. Quality control: manufacturers calculate defect probabilities to set acceptance thresholds. Gambling: casinos design games with specific house edge probabilities. Epidemiology: tracking disease spread probabilities to guide public health decisions. Machine learning: algorithms assign probabilities to classifications.

Lottery probability uses combinations. For a typical 6/49 lottery (pick 6 numbers from 1–49): total combinations = C(49,6) = 13,983,816. Probability of jackpot = 1/13,983,816 ≈ 0.0000000715 or about 1 in 14 million. For Powerball (choose 5 from 69, plus 1 Powerball from 26): C(69,5) × 26 = 11,238,513 × 26 = 292,201,338. Jackpot probability = 1 in ~292 million. You are about 20,000 times more likely to be struck by lightning in a year than to win Powerball in a single ticket.