Probability Calculator

Calculate exact probability for single events, compound events (AND / OR), conditional probability P(A|B), complements, and the probability of at least one success across repeated trials.

Probability Mode

Live calculation
Classic Examples:

Probability is always bounded between 0.0 (impossible) and 1.0 (certain). Odds represent the ratio of favorable to unfavorable chances.

Calculated Probability
16.67%
Fractional Chance1/6
0% (Impossible)16.67%100% (Certain)
Decimal Value0.166667
Odds For1 : 5.00
Odds Against5.00 : 1
Complement P'83.33%

Mathematical Working

• Formula: P(A) = Favorable Outcomes ÷ Total Possible Outcomes

• Substitution: P(A) = 1 ÷ 6 = 0.1667

Core Axioms and Rules of Probability

Probability theory evaluates the mathematical certainty of events on a normalized interval [0, 1]. The foundational rules govern how individual probabilities combine for joint, disjoint, and conditional outcomes.

1. Addition Rule (Union P(A ∪ B))
P(A ∪ B)=P(A) + P(B) − P(A ∩ B)
2. Multiplication Rule (Intersection P(A ∩ B))
P(A ∩ B)=P(A) × P(B|A)=P(A) × P(B)(if independent)
3. Complement & "At Least One" Rule
P(≥ 1 success in n trials)=1 − (1 − p)ⁿ
Step-by-Step Calculation Breakdown (Example: Rolling at least one 6 in 4 rolls)
Step 1: Determine Probability of Failure per Trial (1 − p)
• Success p = 1/6 ≈ 0.1667
• Failure (not rolling a 6) q = 1 − 1/6 = 5/6 ≈ 0.8333
Step 2: Probability of 0 Successes in 4 Trials (q⁴)
• P(0 sixes) = (5/6)⁴ = 625 / 1296 ≈ 0.4823 (48.23%)
Step 3: Subtract from 1 for Complement (At Least One)
• P(≥ 1 six in 4 rolls) = 1 − 0.4823 = 0.5177 (51.77%)

Probability vs Odds Reference Table

Event DescriptionProbability PPercentageOdds ForOdds Against
Coin Flip (Heads)0.500050.0%1 : 1 (Even)1 : 1
Single Die (Roll 6)0.166716.67%1 : 55 : 1
Draw an Ace (Standard Deck)0.07697.69%1 : 1212 : 1
Draw a Spade0.250025.0%1 : 33 : 1
Rolling Pair of Sixes (2 Dice)0.02782.78%1 : 3535 : 1

Frequently Asked Questions

What is probability and how is it defined mathematically?
Probability measures the likelihood of an event occurring on a real scale from 0 (impossible event) to 1 (absolute certainty). In classical probability with equally likely outcomes: P(A) = favorable outcomes / total outcomes.
What is the difference between independent and dependent events?
Events are independent if the occurrence of one does not change the probability of the other (e.g. consecutive coin flips, rolling dice). For independent events, P(A ∩ B) = P(A) × P(B). Events are dependent if one affects the other (e.g. drawing cards without replacement).
What is the addition rule for P(A or B)?
The union rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). We subtract the intersection P(A ∩ B) so that outcomes belonging to both sets are not counted twice. If events are mutually exclusive (cannot happen together), P(A ∩ B) = 0.
What is conditional probability P(A|B)?
Conditional probability measures the probability of event A occurring given that event B has already occurred: P(A|B) = P(A ∩ B) / P(B).
How do you calculate the probability of 'at least one' occurrence?
Using the complement rule: P(at least 1 success in n trials) = 1 − P(zero successes in n trials) = 1 − (1 − p)ⁿ.

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