Probability Calculator

Calculate single, compound (A and B), union (A or B), conditional (A | B) probabilities, and exact odds step-by-step.

Math Audited
Probability of Event A — P(A)0.40 (40.00%)
Probability of Event B — P(B)0.50 (50.00%)
Union P(A or B)
At least one event happens
70.00%
Decimal: 0.7000
Intersection P(A and B)
Both A and B happen simultaneously
20.00%
Decimal: 0.2000
Conditional P(A given B)
Likelihood of A assuming B occurred
40.00%
P(A ∩ B) / P(B)
Neither Event P(Neither)
Neither A nor B occurs (1 - Union)
30.00%
1 - P(A ∪ B)
Event Probability Distribution Visualizer
Event A — P(A)40.00%
Event B — P(B)50.00%
Intersection — P(A and B)20.00%
Union — P(A or B)70.00%
Neither — P(Neither)30.00%
Venn Set TopologyABA∩B
1. Joint Intersection P(A and B):
P(A ∩ B) = P(A) × P(B) = 0.4000 × 0.5000 = 0.2000 (20.00%)
2. General Addition Rule (Union P(A or B)):
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4000 + 0.5000 - 0.2000 = 0.7000 (70.00%)
3. Conditional Probability P(A|B):
P(A|B) = P(A ∩ B) / P(B) = 0.2000 / 0.5000 = 0.4000 (40.00%)
4. Conditional Probability P(B|A):
P(B|A) = P(A ∩ B) / P(A) = 0.2000 / 0.4000 = 0.5000 (50.00%)
Scientific References & Assumptions
Core Assumptions & Axioms:
  • Kolmogorov's Axioms: All probabilities exist in the closed interval [0, 1] with the probability of the entire sample space equal to 1.
  • In independent mode, the occurrence of event A yields zero information regarding event B, enforcing P(A ∩ B) = P(A) × P(B).
  • In mutually exclusive mode, events are disjoint (P(A ∩ B) = 0).
  • Fréchet-Hoeffding Bounds: The intersection cannot exceed the smaller of P(A) and P(B), and must be large enough to prevent the union from exceeding 100%.
Primary Citations:
  • Ross, S. M. (2019). A First Course in Probability (9th Edition). Pearson Education.
  • NIST / SEMATECH (2026). e-Handbook of Statistical Methods. National Institute of Standards and Technology.

About the Probability Calculator

Easily compute single-event, compound, conditional, and independent probabilities using standard mathematical axioms. Toggle between independent and mutually exclusive events, convert between odds and probability, and view step-by-step intermediate formulas.

Mathematical Formula & Logic

Probability quantifies the exact likelihood of an event occurring on a scale from 0 to 1 (or 0% to 100%): 1. Classical Single Event Probability: P(A) = N(A) / N(S) Where N(A) is the number of favorable outcomes and N(S) is the total number of equally likely outcomes in the sample space S. 2. Complementary Event Probability: P(A') = 1 - P(A) The probability that event A does not occur. 3. General Addition Rule (Union of Two Events): P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Calculates the probability that at least one of the two events happens. For mutually exclusive events, P(A ∩ B) = 0, reducing the union to P(A) + P(B). 4. Multiplication Rule (Intersection of Independent Events): P(A ∩ B) = P(A) × P(B) Calculates the joint likelihood of both events occurring simultaneously when they are statistically independent. 5. Conditional Probability: P(A | B) = P(A ∩ B) / P(B) The probability of event A occurring given that conditioning event B has already occurred. 6. Odds in Favor & Odds Against: Odds in Favor = P(A) / (1 - P(A)) Odds Against = (1 - P(A)) / P(A) To convert odds of a : b in favor to probability: P(A) = a / (a + b).

Step-by-Step Example

Worked Examples: 1. Compound Independent Events (P(A) = 0.40, P(B) = 0.50): Let Event A be a 40% chance of rain on Saturday and Event B be a 50% chance of rain on Sunday, assuming both days are independent. - Joint Intersection P(A and B) = 0.40 × 0.50 = 0.20 (20% chance of rain on both days). - Union P(A or B) = P(A) + P(B) - P(A and B) = 0.40 + 0.50 - 0.20 = 0.70 (70% chance of rain on at least one day). - Neither Event P(Neither) = 1 - P(A or B) = 1 - 0.70 = 0.30 (30% chance of a completely dry weekend). 2. Odds to Probability Conversion (Odds 3 : 1 in Favor): Suppose a team has 3 to 1 odds in favor of winning a championship match. - Total sample space parts = 3 favorable + 1 unfavorable = 4 total parts. - Exact Probability P(Win) = 3 / 4 = 0.75 (75% winning probability). - Complementary Probability P(Loss) = 1 - 0.75 = 0.25 (25% losing probability). - Odds Against winning are exactly 1 : 3.

Reference Data & Values

probability metricformulaapplicabilityexample value
Classical Probability P(A)N(A) / N(S)Finite sample space with equally likely discrete outcomes3 / 12 = 25%
Complementary Event P(A')1 - P(A)Exact logical opposite of event A1 - 0.25 = 75%
Union P(A or B)P(A) + P(B) - P(A and B)Probability that at least one of two events occurs0.4 + 0.5 - 0.2 = 70%
Intersection P(A and B)P(A) × P(B)Joint occurrence of two statistically independent events0.4 × 0.5 = 20%
Conditional P(A | B)P(A and B) / P(B)Likelihood of A assuming conditioning event B occurred (P(B) > 0)0.2 / 0.5 = 40%

Frequently Asked Questions

Independent events do not affect each other's likelihood of occurring (e.g., flipping two coins). Mutually exclusive events cannot happen at the same time (e.g., rolling a 2 and a 5 on a single die). If two events with non-zero probabilities are mutually exclusive, they are strongly dependent, because the occurrence of one guarantees the other does not happen.
When you sum P(A) and P(B), the overlapping outcomes where both events occur (the intersection) get added twice. Subtracting P(A and B) once corrects for this double-counting, giving the exact probability of the union.
Conditional probability P(A | B) measures the likelihood of event A occurring under the assumption or knowledge that event B has already occurred. Mathematically, it restricts the sample space to B and divides the joint intersection P(A and B) by P(B).
If the odds in favor of an event are a to b (a : b), the probability is calculated as a / (a + b). For example, odds of 4 to 1 in favor correspond to a probability of 4 / (4 + 1) = 0.80 (80%).
No. By Kolmogorov's fundamental axioms of probability, all probabilities must be real numbers between 0 (representing absolute impossibility) and 1 (representing absolute certainty).
If event B has a probability of zero (an impossible event), conditional probability P(A | B) is mathematically undefined because division by zero is impossible.
Fréchet-Hoeffding bounds state that the joint probability P(A and B) can never exceed the smaller of the two individual probabilities, and must be large enough that the union P(A or B) does not exceed 100%.