Probability Calculator

Calculate Probability for Single, Independent & Conditional Events

✅ Favorable Outcomes

🎲 Total Outcomes

P(A) — Probability of Event A

P(B) — Probability of Event B

🔗 Relationship

P(A ∩ B) — Probability of both A and B

P(B) — Probability of B

Result

📝 Step-by-step solution

Select calculation type and enter values

💡 Probability is always between 0 and 1 | 0 = impossible | 1 = certain

📖 What is Probability?

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Probability is widely used in statistics, gambling, weather forecasting, insurance, machine learning, and everyday decision-making.

🧮 Probability Formula (Single Event)

P(A) = Number of Favorable Outcomes / Total Number of Outcomes

🧮 Probability Formula (Two Events)

Independent events: P(A ∩ B) = P(A) × P(B)
Mutually exclusive: P(A ∪ B) = P(A) + P(B)
General union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Conditional probability: P(A | B) = P(A ∩ B) / P(B)

📝 Worked Examples

Example 1: Rolling a Die — Probability of Getting an Even Number

Total outcomes = 6 (1, 2, 3, 4, 5, 6)
Favorable outcomes = 3 (2, 4, 6)
P(even) = 3 / 6 = 0.5 = 50%

Example 2: Two Coin Flips — Probability of Two Heads

P(Head) = 0.5 for each flip (independent events)
P(Head ∩ Head) = 0.5 × 0.5 = 0.25 = 25%

Example 3: Conditional — Probability of Rain Given Cloudy Sky

P(Rain ∩ Cloudy) = 0.30
P(Cloudy) = 0.50
P(Rain | Cloudy) = 0.30 / 0.50 = 0.60 = 60%

💡 Common Probability Values to Remember

Coin flip: P(Heads) = 1/2 = 0.50
Die roll (specific number): P(1) = 1/6 ≈ 0.1667
Card draw (Ace): P(Ace) = 4/52 ≈ 0.0769
Two heads in a row: 0.50 × 0.50 = 0.25
Birthday match (23 people): ≈ 0.507

⚠️ Limitations & Assumptions

📌 Important Notes

Assumes events are equally likely for single-event calculations.
For two-event mode, you must correctly identify whether events are independent, mutually exclusive, or general union.
Conditional probability requires P(B) > 0.
Does not handle dependent events without explicit joint probability input.
Results are mathematical probabilities — real-world outcomes may vary.

📚 Sources & References

Authoritative Sources

Khan Academy — Probability and Statistics
OpenStax — Introductory Statistics
MIT OpenCourseWare — Probabilistic Systems Analysis
Ross, S. — A First Course in Probability

❓ Frequently Asked Questions

What does probability 0.5 mean?

A probability of 0.5 means the event has a 50% chance of occurring — equal odds of happening or not happening (like a fair coin flip).

What is the difference between independent and mutually exclusive events?

Independent events can occur simultaneously, and one event doesn't affect the other (e.g., two coin flips). Mutually exclusive events cannot occur at the same time (e.g., rolling a 1 and a 2 on a single die).

Can probability be greater than 1?

No — probability is always between 0 and 1 (or 0% and 100%). A value greater than 1 means there's an error in the calculation.

What is conditional probability?

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. Formula: P(A|B) = P(A ∩ B) / P(B).

Are the results accurate?

Yes — the calculator uses standard probability formulas. However, real-world data may deviate from theoretical probabilities due to sampling error or biased conditions.

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⚠️ Disclaimer

This Probability Calculator is provided for informational and educational purposes only. Results are based on standard probability theory and assume equally likely outcomes. Real-world events may not always follow theoretical probabilities. Always verify with a qualified statistician for critical decisions.