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A sample space is the set of all possible outcomes of a random experiment, typically denoted by S or Ω. There is no single universal formula for sample spaces; instead, the method for determining the size and structure of a sample space depends on the type of experiment being conducted. For experiments with multiple stages, the multiplication rule applies: the total number of outcomes equals the product of outcomes at each stage. For example, rolling two six-sided dice produces 6 × 6 = 36 possible ordered pairs, from (1,1) to (6,6). This formula generalizes to n₁ × n₂ ×... × nₖ for k independent stages. For coin tosses specifically, the sample space size follows the formula 2ⁿ, where n is the number of coins. Three coin tosses yield 8 outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. When order does not matter, combinatorics formulas become necessary. The combination formula C(n,r) = n! / (r!(n-r)!) calculates the number of ways to choose r items from n without regard to order, as in selecting 3 cards from a 52-card deck, which produces 22,100 possible combinations. For continuous sample spaces, such as selecting a real number between 0 and 1, the sample space is infinite and cannot be enumerated. In these cases, probability density functions replace discrete counting methods, and probabilities are computed through integration rather than summation. Tree diagrams provide a visual method for constructing sample spaces, especially for multi-stage experiments. Each branch represents a possible outcome at that stage, and following a complete path from root to leaf yields one element of the sample space. The probability of any outcome equals the product of probabilities along its path. Once the sample space is established, the probability of an event E is calculated as P(E) = P(e₁) + P(e₂) +... + P(eₖ), where e₁ through eₖ are the individual outcomes comprising the event. For equally likely outcomes, this simplifies to the familiar ratio: P(E) = (number of favorable outcomes) / (total outcomes in sample space).