🎲 Probability Calculator

Probability is expressed as a fraction or decimal between 0 and 1, representing the likelihood of an event occurring. Odds compare the number of favorable outcomes to unfavorable outcomes, expressed as a ratio (e.g., 3:2). You can convert between them: if probability is p, then odds are p:(1-p).

Two events are mutually exclusive if they cannot occur at the same time. For example, when rolling a die, getting a 3 and getting a 5 are mutually exclusive. For mutually exclusive events, P(A or B) = P(A) + P(B). If events are not mutually exclusive, you must subtract P(A and B) to avoid double-counting.

Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin twice - the first flip doesn't affect the second. For independent events, P(A and B) = P(A) × P(B). If events are dependent, you need to use conditional probability.

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. The formula is P(A|B) = P(A and B) / P(B). For example, if you draw a card from a deck and it's red, the probability it's a heart is P(Heart|Red) = P(Heart and Red) / P(Red) = 0.25 / 0.5 = 0.5.

The complement rule states that the probability of an event NOT occurring is 1 minus the probability that it does occur: P(not A) = 1 - P(A). This is useful when it's easier to calculate the probability of the opposite event. For example, it's easier to find the probability of at least one success by calculating 1 - P(all failures).

No, probability values must be between 0 and 1 (or 0% to 100%). A probability of 0 means the event is impossible, 1 means it's certain, and values in between represent varying degrees of likelihood. If you calculate a probability greater than 1, there's an error in your calculation or assumptions.

Bayes' Theorem is a formula for calculating conditional probabilities in reverse: P(A|B) = P(B|A) × P(A) / P(B). It's used to update probabilities based on new evidence and is fundamental in statistics, machine learning, and decision-making. It's particularly useful in medical testing, spam filtering, and risk assessment.