Probability Calculator — Single & Multiple Events, Bayes' Theorem & Odds

Probability of Two Events

Probability quantifies the likelihood of a specific event occurring within a defined sample space of possible outcomes. Expressed as a real number on the closed interval [0, 1], a probability of 0 indicates an impossible outcome, whereas 1 represents absolute certainty. In classical probability theory, the probability of an event E is determined by the ratio of favorable outcomes to the total number of equiprobable outcomes in the sample space:

When analyzing two distinct events, A and B, the relationship between them depends heavily on whether they are independent (the occurrence of one does not alter the likelihood of the other), mutually exclusive (they cannot occur simultaneously), or conditionally dependent. The calculator above evaluates all primary joint set operations for two independent events, including complementary probabilities, intersections, inclusive unions, symmetric differences (exclusive OR), and un-occurred complements.

Complement of A and B

The complement of an event A, denoted as P(A') or P(Aᶜ), represents the probability that event A does NOT occur. By the fundamental law of total probability, the sum of an event and its complement across the entire sample space must equal 1:

For instance, if a manufacturing line has a quality defect probability of P(A) = 0.04 (a 4% defect rate), the probability that a randomly sampled unit is non-defective is calculated as:

Similarly, the complement of event B is computed as P(B') = 1 - P(B). It is important to recognize that unless A and B form a complete two-set partition of the sample space, P(A) and P(B) operate independently, so P(A') and P(B') measure separate non-occurrence probabilities.

Intersection of A and B

The intersection of two events A and B, written as P(A ∩ B) or P(A AND B), evaluates the joint probability that both event A and event B occur concurrently. If A and B are mutually exclusive, they share no sample points, rendering P(A ∩ B) = 0 (such as rolling a single die and landing on both 2 and 5 simultaneously).

When events A and B are statistically independent, the joint probability simplifies to the product of their individual probabilities:

For dependent events, conditional probability governs the outcome: P(A ∩ B) = P(A) × P(B|A), where P(B|A) represents the probability of B occurring given that A has already transpired. Consider an urn containing 12 balls: 8 red and 4 blue. If two balls are drawn sequentially without replacement:

Union of A and B

The union of two events, denoted by P(A ∪ B) or P(A OR B), measures the probability that at least one of the events occurs. In standard probability theory, this represents the inclusive OR condition—meaning event A occurs, event B occurs, or both occur simultaneously.

For mutually exclusive events where no overlap exists:

For non-mutually exclusive events, simply adding P(A) and P(B) double-counts the overlapping intersection region. To correct for this, the general Principle of Inclusion-Exclusion is applied:

For example, when rolling a fair 6-sided die (sample space S = {1,2,3,4,5,6}), let event A be rolling an even number (P(A) = 3/6) and event B be rolling a number greater than 3 (P(B) = 3/6 = {4,5,6}). The intersection is {4,6} (P(A ∩ B) = 2/6). The union probability is calculated as:

Exclusive OR of A and B

The Exclusive OR operation, denoted as P(A Δ B) or P(A XOR B), measures the probability that exactly one of the two events occurs, excluding the possibility of both occurring simultaneously. Mathematically, it subtracts twice the joint intersection from the sum of the individual probabilities:

Consider a dual-server cloud infrastructure where Server A has an uptime probability of P(A) = 0.95 and Server B has an uptime probability of P(B) = 0.90. Assuming independent operation, the joint uptime is P(A ∩ B) = 0.95 × 0.90 = 0.855. The probability that exactly one server is running while the other is down is:

Probability of a Series of Independent Events

Evaluating repeated trials of independent events involves exponential power laws. If event A has a single-trial probability p and is repeated n times, and event B has a single-trial probability q and is repeated m times, the compound outcomes follow distinct binomial probability rules:

• All n trials of A succeed: P(Aⁿ) = pⁿ
• Zero trials of A succeed: P(A')ⁿ = (1 - p)ⁿ
• At least one trial of A succeeds: 1 - (1 - p)ⁿ
• Both A succeeds n times and B succeeds m times: pⁿ × q^m
• Neither series succeeds: (1 - p)ⁿ × (1 - q)^m

Normal Distribution

The normal distribution, or Gaussian distribution, is a continuous probability distribution characterized by a symmetric, bell-shaped curve defined by its mean (μ) and variance (σ²). The probability density function (PDF) is given by:

In standard normal form where μ = 0 and σ = 1, raw data values X are transformed into standard Z-scores representing the distance from the mean in units of standard deviation:

According to the empirical rule (68-95-99.7 rule), approximately 68.27% of all observations fall within 1 standard deviation of the mean (-1 ≤ Z ≤ 1), 95.45% fall within 2 standard deviations (-2 ≤ Z ≤ 2), and 99.73% fall within 3 standard deviations (-3 ≤ Z ≤ 3).