Probability Calculator — Free Online Probability Tool

How to Use the Probability Calculator

1. Select the calculation type: Choose from five modes. "Single Event" calculates P(A) from favorable and total outcomes. "Both Events" finds P(A and B). "Either Event" finds P(A or B). "Complement" finds P(not A). "Conditional" finds P(A|B).
2. Enter values: For single events, enter the number of favorable outcomes and total outcomes. For combined and conditional modes, enter probability values between 0 and 1. For "Both Events (AND)" mode, specify whether events are independent.
3. Check the independence setting: When calculating P(A and B), checking "Events are independent" applies the multiplication rule P(A) × P(B). Unchecking it requires you to enter P(A and B) directly, useful for dependent events.
4. Read the results: The probability appears as a decimal, percentage, and (for single events) as odds. The formula used is shown for transparency. The complement (probability of the event NOT happening) is always displayed.

All calculations update in real time. Switch between modes to explore different probability scenarios. The calculator ensures probabilities stay within the valid 0 to 1 range.

Probability Formulas

Variables Explained

• P(A): The probability of event A occurring, ranging from 0 (impossible) to 1 (certain).
• P(A ∩ B): The probability of both A and B occurring (intersection).
• P(A ∪ B): The probability of either A or B (or both) occurring (union).
• P(A'): The probability of A not occurring (complement). Always equals 1 - P(A).
• P(A|B): The probability of A occurring given that B has occurred (conditional probability).

Step-by-Step Example

A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a red or blue marble?

1. Count favorable outcomes: Red = 5, Blue = 3, Total favorable = 8
2. Count total outcomes: 5 + 3 + 2 = 10
3. Since drawing red and blue are mutually exclusive: P(Red or Blue) = P(Red) + P(Blue)
4. P(Red) = 5/10 = 0.5, P(Blue) = 3/10 = 0.3
5. P(Red or Blue) = 0.5 + 0.3 = 0.8 (80%)

There is an 80% chance of drawing either a red or blue marble. The complement is P(Green) = 0.2 or 20%. You can also verify: 8 favorable / 10 total = 0.8.

Practical Examples

Example 1: Amy's Weather Planning

Amy is planning an outdoor wedding. Historical data shows a 30% chance of rain on any given day in June and a 20% chance of high winds. She assumes these events are independent. She wants to know the probability of having good weather (neither rain nor high wind).

• P(Rain) = 0.30, P(Wind) = 0.20
• P(No Rain) = 1 - 0.30 = 0.70
• P(No Wind) = 1 - 0.20 = 0.80
• P(No Rain AND No Wind) = 0.70 × 0.80 = 0.56 (56%)

Amy has a 56% chance of good weather. She also calculates the probability of at least one problem: P(Rain or Wind) = 0.30 + 0.20 - (0.30 × 0.20) = 0.44 (44%). She decides to book an indoor backup venue given the 44% chance of weather issues.

Example 2: Carlos's Quality Control Analysis

Carlos inspects electronic components where 2% have a soldering defect and 3% have a component defect. These defects are independent. He wants to know the probability that a random unit has at least one defect.

• P(Solder) = 0.02, P(Component) = 0.03
• P(Both) = 0.02 × 0.03 = 0.0006
• P(At least one) = 0.02 + 0.03 - 0.0006 = 0.0494 (4.94%)
• P(No defects) = 1 - 0.0494 = 0.9506 (95.06%)

About 4.94% of units have at least one defect. For a batch of 1,000 units, Carlos expects approximately 49 defective units. If the company needs 99% defect-free output, they must improve both processes significantly. He can use our sample size calculator to determine how many units to inspect for quality assurance.

Example 3: Dr. Patel's Medical Screening

Dr. Patel uses a screening test with 95% sensitivity (P(Positive|Disease) = 0.95) and 90% specificity (P(Negative|No Disease) = 0.90). The disease prevalence is 1% (P(Disease) = 0.01). She wants to find the probability that a patient with a positive result actually has the disease.

• P(Positive and Disease) = 0.95 × 0.01 = 0.0095
• P(Positive and No Disease) = 0.10 × 0.99 = 0.099
• P(Positive) = 0.0095 + 0.099 = 0.1085
• P(Disease|Positive) = 0.0095 / 0.1085 = 0.0876 (8.76%)

Despite the test being 95% accurate, a positive result only means an 8.76% chance of actually having the disease. This counterintuitive result (Bayes' theorem in action) occurs because the disease is rare — most positive results are false positives from the 99% of healthy people. This is why medical screening often requires confirmatory tests.

Common Probability Reference Table

Tips and Complete Guide

The Complement Strategy

When asked "what is the probability of at least one success?", it is usually easier to calculate the probability of zero successes and subtract from 1. For example, the probability of getting at least one 6 in four die rolls: P(at least one 6) = 1 - P(no 6 in any roll) = 1 - (5/6)⁴ = 1 - 0.482 = 0.518. This approach avoids counting all the individual success cases (exactly 1, exactly 2, exactly 3, or exactly 4 sixes).

Understanding Mutually Exclusive vs. Independent

These terms are often confused. Mutually exclusive events cannot happen simultaneously (rolling a 3 AND a 5 on one die). Independent events do not influence each other's probabilities (rolling a 3 on the first die and a 5 on the second). Mutually exclusive events are NOT independent (if A happens, B definitely cannot happen, so knowing A tells you about B). For mutually exclusive events, P(A and B) = 0. For independent events, P(A and B) = P(A) × P(B).

Probability in Decision Making

Expected value combines probability with outcomes: EV = Σ(probability × outcome). For example, if a game costs $5 with a 10% chance of winning $40 and 90% chance of losing, EV = 0.10 × $40 + 0.90 × (-$5) = $4 - $4.50 = -$0.50. The negative expected value means you lose $0.50 on average per game. This framework applies to investment decisions, insurance, business strategy, and any situation involving risk. For counting methods used in probability, try our permutation and combination calculator.

Common Mistakes to Avoid

• Adding probabilities for non-mutually-exclusive events: P(Heart or King) ≠ P(Heart) + P(King) because the King of Hearts is counted twice. You must subtract P(Heart AND King) = 1/52: P = 13/52 + 4/52 - 1/52 = 16/52.
• Assuming independence without justification: Multiplying P(A) × P(B) is only valid for independent events. If events are related (like weather conditions on consecutive days), you need conditional probabilities or joint probability data.
• The Gambler's Fallacy: After 10 coin flips showing heads, the probability of the next flip being heads is still 50%. Previous outcomes do not influence future independent events. The coin does not "owe" you tails.
• Confusing "at least one" with "exactly one": P(at least one head in 3 flips) = 7/8, but P(exactly one head) = 3/8. "At least one" includes one, two, and three heads. Always clarify which probability you need.
• Forgetting the base rate: In medical testing and other screening scenarios, the base rate (prevalence) dramatically affects the interpretation of positive results. A positive test for a rare disease is much more likely to be a false positive than a true positive, as shown in our Bayes' theorem example above.

Frequently Asked Questions