If you flip a coin and roll a die, what are the odds of heads and a six? The Probability Calculator answers questions like this for two independent events. You enter two probabilities, and it returns five combined results.
It is built for students, teachers, and anyone checking a quick "this and that" chance. It does not model dice, cards, or distributions. It works only with the two numbers you type.
Table of Contents
- What the Calculator Does
- How to Use It, Step by Step
- The Probability Formulas
- Worked Example
- How to Read the Results
- What Independent Means
- Limitations and Assumptions
- FAQs About the Probability Calculator
What the Calculator Does
The tool takes two inputs and shows five results. All of them assume the two events do not affect each other, and a note under the inputs says so.
- P(A and B) [Intersection]: the chance that both events happen.
- P(A or B) [Union]: the chance that at least one event happens.
- P(Not A) [Complement]: the chance that event A does not happen.
- P(Not B) [Complement]: the chance that event B does not happen.
- P(A XOR B) [Exclusive Or]: the chance that exactly one of the two events happens.
Each result appears as a decimal with four places and as a percent with one place. The results card is already filled in when the page loads.
How to Use It, Step by Step
- Type a number from 0 to 1 into Probability of Event A: P(A). The starting value is 0.5.
- Type a number from 0 to 1 into Probability of Event B: P(B). The starting value is 0.3.
- Press Calculate Probabilities. The cards also update as you type and on Enter.
- Read the five cards under the heading "Compound Event Analysis (Independent Events)".
Use decimals, not percents. A 25% chance is entered as 0.25, and any decimal such as 0.37 is accepted.
The Probability Formulas
For independent events, the chance of both is the product of the two chances. The other results build on that product.
P(A)×P(B)=P(A and B)
- P(A and B) = P(A) × P(B).
- P(A or B) = P(A) + P(B) - P(A and B).
- P(Not A) = 1 - P(A).
- P(Not B) = 1 - P(B).
- P(A XOR B) = P(A) + P(B) - 2 × P(A and B).
The union formula subtracts the overlap once, because the "both" case is counted in each event. The XOR formula subtracts it twice, so the "both" case is removed completely.
Worked Example
Tomas coaches a youth soccer team. His team has a 0.35 chance to win Saturday's game and a 0.6 chance to win Sunday's game. He treats the two games as independent.
| Result | Calculation | Output |
|---|
| P(A and B) | 0.35 × 0.6 | 0.2100 (21.0%) |
| P(A or B) | 0.35 + 0.6 - 0.21 | 0.7400 (74.0%) |
| P(Not A) | 1 - 0.35 | 0.6500 (65.0%) |
| P(Not B) | 1 - 0.6 | 0.4000 (40.0%) |
| P(A XOR B) | 0.35 + 0.6 - 2 × 0.21 | 0.5300 (53.0%) |
So Tomas has a 21% chance of a weekend sweep and a 74% chance of at least one win. With the starting values of 0.5 and 0.3, the cards show 0.1500, 0.6500, 0.5000, 0.7000, and 0.5000.
How to Read the Results
P(A and B) is always less than or equal to the smaller input. Two unlikely events together are even less likely.
P(A or B) is always at least as large as the bigger input. It can never fall below either single chance.
P(Not A) and P(Not B) each look at one event only. They are the chance that event does not happen.
P(A XOR B) is the union minus the overlap. Check it by adding "A only" and "B only". That is 0.35 × 0.4 + 0.65 × 0.6, or 0.14 + 0.39, which is 0.53.
What Independent Means
Two events are independent when one outcome does not change the odds of the other. A coin flip and a die roll are independent. Two draws from a deck without replacement are not.
If your events are linked, the product rule gives the wrong answer. Rain today and rain tomorrow often depend on each other, so treat that pair with care.
For counting-based problems, the combination calculator and the permutation calculator count outcomes first. You can then divide by the total to get a chance.
For repeated trials with the same success chance, use the binomial probability calculator. For bell-curve questions, try the normal distribution calculator.
To turn a decimal result into a fraction, the decimal to fraction calculator helps. More tools live in the math and science calculators hub.
Limitations and Assumptions
- The tool assumes the two events are independent. It has no setting for dependent or mutually exclusive events.
- Each box must hold a number from 0 to 1. A blank, non-numeric, or out-of-range box shows a red message naming the field, and every card shows N/A.
- A typed 0 is valid and is not treated as blank.
- Results are rounded half up to four decimals, and percents to one decimal. So a result of 0.9999 shows as 0.9999 (100.0%).
- The tool does not check that your input probabilities are realistic. It only does the arithmetic.
- It is a teaching aid, not a risk model for money, health, or safety decisions.
FAQs About the Probability Calculator
What is the probability of two independent events both happening?
Multiply the two chances. For 0.5 and 0.3, the chance of both is 0.15, or 15%.
What is the difference between "or" and "XOR" here?
"Or" counts every case where at least one event happens, including both. XOR counts only the cases where exactly one event happens.
Can I enter percents instead of decimals?
No. Enter 0.25 for 25%. A value above 1, up to 100, adds a hint to divide by 100, so 50% is 0.5.
Why does P(A or B) not simply add the two chances?
Adding counts the "both" case twice. The tool subtracts P(A and B) once to fix that.
Does this work for events that depend on each other?
No. The formulas assume independence. For linked events you need a conditional probability, which this tool does not compute.
What do probabilities of 0 and 1 mean here?
A value of 0 means the event never happens, and 1 means it always happens. With 1 and 1, both complement cards show 0.0000 (0.0%).
This calculator gives general estimates only and is not professional advice. Reviewed for general accuracy by Prof. Dr. Khalil Mudassar, PhD.