Probability Calculator - Compound Events, Independent & Odds

Calculate single and compound event probabilities, intersection P(A ∩ B), union P(A ∪ B), complementary probability, conditional probability, and betting odds ratios.

⚙️ Configuration & Inputs

Probability of Event A: P(A) Decimal between 0 and 1 (e.g. 0.5 for coin flip)

Probability of Event B: P(B) Decimal between 0 and 1

Event Relationship

📊 Results & Analysis

Both Occur: P(A ∩ B) 0.1250 (12.50%) P(A and B) Intersection

Either Occurs: P(A ∪ B) 0.6250 (62.50%) P(A or B) Union

Neither Occurs: P(Neither) 0.3750 (37.50%) 1 - P(A ∪ B)

Only A Occurs: P(A and not B) 0.3750 (37.50%) P(A) - P(A ∩ B)

Odds in Favor of Event A 1 : 1 Even Odds (50% implied probability)

Odds in Favor of Event B 1 : 3 3 to 1 against event B

Fundamental Rules of Probability Theory

Probability quantifies the likelihood of an event occurring on a normalized scale from 0.0 (impossible) to 1.0 (certainty).

1. The Addition Rule (Union - "OR")

For any two events A and B:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

If the events are mutually exclusive (they cannot happen simultaneously), $P(A \cap B) = 0$, simplifying to $P(A \cup B) = P(A) + P(B)$.

2. The Multiplication Rule (Intersection - "AND")

If two events are statistically independent (the occurrence of A has zero effect on B):

P(A ∩ B) = P(A) × P(B)

3. Converting Probability into Odds

Odds in favor represent the ratio of successful outcomes to failures: Odds = P(A) / [1 - P(A)].

Frequently Asked Questions

The gambler's fallacy is the mistaken belief that past independent random events influence future events. If a fair coin lands on Heads 5 times in a row, the probability of Heads on the 6th flip remains exactly 50%.

Decimal odds (e.g., 2.50) convert to implied probability via 1 / Odds = 1 / 2.50 = 40%. Fractional odds of 4/1 mean an implied probability of 1 / (4 + 1) = 20%.

Mutually exclusive events cannot happen at the same time. For example, rolling a single die cannot produce both a 2 and a 5 on the exact same roll.