Probability Fraction Calculator
Enter the number of favorable outcomes and total outcomes to get the probability as a reduced fraction, a decimal, and a percentage. Switch to Two Events mode to solve complement, intersection, union, and exclusive-OR probabilities for two independent events. Every result includes a step-by-step breakdown of the arithmetic.
Formula
Worked example
A bag has 12 marbles: 3 red, 5 blue, 4 green. The probability of picking a red marble is 3/12. GCD(3, 12) = 3, so the reduced fraction is 1/4 = 0.25 = 25%. The complement (not red) is 3/4 = 75%. If a second independent draw has P(blue) = 5/12, then P(red AND blue on two draws) = 1/4 x 5/12 = 5/48 (about 10.4%).
What is a probability fraction?
A probability fraction expresses the chance of an event occurring as a ratio of the number of favorable outcomes to the total number of equally likely outcomes. For example, if a bag contains 3 red marbles and 9 other marbles, the probability of drawing red is 3/12, which reduces to 1/4. This fraction form is the most transparent way to communicate probability because it shows exactly what is being compared. The denominator tells you the size of the sample space and the numerator tells you how many of those outcomes satisfy the event.
How to calculate probability as a fraction
The steps are: (1) count the number of outcomes that satisfy the event, called favorable outcomes; (2) count all equally likely outcomes in the sample space, called total outcomes; (3) write the fraction as favorable / total; (4) find the greatest common divisor (GCD) of the two numbers; (5) divide both numerator and denominator by the GCD to get the reduced (simplest) form. For example, 6/8 reduces to 3/4 because GCD(6, 8) = 2. If GCD = 1 the fraction is already in simplest form. Converting to a decimal is then just a division: 3 / 4 = 0.75. To get a percentage, multiply by 100: 75%.
Two-event probabilities - union, intersection, and complement
When two events A and B are independent, four key relationships apply. The intersection (both A and B occur) is P(A) x P(B). The union (at least one of A or B occurs) is P(A) + P(B) - P(A and B), because without subtracting the intersection you would count those outcomes twice. The exclusive-OR (exactly one of A or B occurs) is P(A) + P(B) - 2 x P(A and B). The complement of any event A is P(A') = 1 - P(A). Two events are mutually exclusive if they cannot both occur, in which case P(A and B) = 0 and the union simplifies to P(A) + P(B).
Conditional probability and Bayes theorem
Conditional probability measures the chance of event A given that event B has already occurred, written P(A|B). The formula is P(A|B) = P(A and B) / P(B). This differs from the simple probability P(A) whenever A and B are not independent. If knowing B occurred changes the likelihood of A, the events are dependent. Bayes' theorem extends this idea: P(A|B) = P(B|A) x P(A) / P(B). It is used in spam filters, medical diagnostics, and machine learning to update probabilities as new information arrives. A useful check: if P(A|B) = P(A), the events are independent and conditioning on B reveals nothing new about A.
Probability scale reference
| Fraction | Decimal | Percentage | Verbal description |
|---|---|---|---|
| 0/1 | 0.0000 | 0% | Impossible |
| 1/20 | 0.0500 | 5% | Very unlikely |
| 1/10 | 0.1000 | 10% | Unlikely |
| 1/4 | 0.2500 | 25% | Unlikely |
| 1/3 | 0.3333 | 33.3% | Less likely than not |
| 1/2 | 0.5000 | 50% | Even odds |
| 2/3 | 0.6667 | 66.7% | More likely than not |
| 3/4 | 0.7500 | 75% | Likely |
| 9/10 | 0.9000 | 90% | Very likely |
| 1/1 | 1.0000 | 100% | Certain |
Common probability values and their fraction, decimal, and verbal equivalents.
Frequently asked questions
How do I simplify a probability fraction?
Find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it. For example, 8/12 has GCD = 4, giving 2/3. If the GCD is 1, the fraction is already in its simplest form. This calculator performs the reduction automatically.
Can a probability fraction be greater than 1?
No. Probability is always between 0 and 1 inclusive. A fraction greater than 1 would mean there are more favorable outcomes than total outcomes, which is not possible. If you see a fraction greater than 1, check that you have entered favorable and total outcomes correctly.
What does a probability of 0 or 1 mean?
A probability of 0 means the event is impossible - it can never occur. A probability of 1 means the event is certain - it always occurs. All real-world events have probabilities strictly between 0 and 1.
What is the difference between odds and probability?
Probability is expressed as favorable / total (e.g., 1/4 = 25%). Odds are expressed as favorable : unfavorable (e.g., 1:3). To convert from probability p to odds in favor, compute p / (1 - p). To convert odds a:b to probability, compute a / (a + b). Sports and gambling often use odds, while science and statistics use probability fractions.
How do I find the probability that two events both occur?
If the events are independent (one does not affect the other), multiply their individual probabilities: P(A and B) = P(A) x P(B). For example, if P(A) = 0.5 and P(B) = 0.3, then P(A and B) = 0.15. If the events are dependent, you need the conditional probability: P(A and B) = P(A|B) x P(B), which is what the Conditional mode of this calculator handles.
What does complement mean in probability?
The complement of an event A, written A' or not-A, is the event that A does NOT occur. Its probability is always 1 - P(A). If the probability of rain tomorrow is 0.3, the probability of no rain is 0.7. Complements are useful because it is sometimes easier to calculate the probability of failure and subtract from 1.