From the probabilities of two events that do not affect each other (independent events), this finds the probability that both happen, that at least one happens, and more. Enter each probability as a number from 0 to 1 (for example, 30% is 0.3).
Enter each probability as a number from 0 to 1 (for example, 30% is 0.3).
Result and figure
Enter the two probabilities in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the probability of each of two events, and you get the probability that both happen, \(P(A \cap B)\), and that at least one happens, \(P(A \cup B)\), on the spot
- It also calculates the probability that exactly one happens, \(P(A \Delta B)\), that neither happens, that only A happens, and more, all at once
- Questions like "There is a 50% chance of sun on day 1 of my trip and a 30% chance on day 2. What is the chance of sun on both days?" are solved in one step
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This works only when the two events do not affect each other (they are independent). A die and a coin, where one result has nothing to do with the other, are fine. It does not work when one result changes the probability of the other, as in "drawing two tickets without putting the first one back".
What is this calculation used for?
Estimating the chance of good weather on both days of a trip or event
If there is a 50% chance of sun on day 1 and a 30% chance on day 2, the chance of sun on both days is only \(0.5 \times 0.3 = 0.15\) (15%). (This is a rough estimate that treats the two days' weather as independent.)
Because you multiply the probabilities, the chance that "both go well" is often much smaller than you expect. Having a feel for this number changes how you make plans.
The chance of getting into at least one of two schools (college admissions, jobs, lotteries)
If you apply to your first-choice college with a 50% chance of admission and to a second college with a 30% chance, the chance of getting into at least one is \(0.5 + 0.3 - 0.15 = 0.65\) (65%). (This assumes the two decisions are independent.)
For a big life choice, you can check with numbers, not just gut feeling, how much adding another option raises your overall chance of success.
Checking how safe your backups are (data and emergency preparedness)
If you store the same data on two disks that each have a 1% chance of failing in a year, the chance of losing both at once drops to \(0.01 \times 0.01 = 0.0001\) (1 in 10,000), as long as the failures are independent.
"Double your safeguards and the risk shrinks by multiplication." This idea is the basis of every kind of preparation, from backing up your photos to storing emergency supplies in more than one place.
Understanding medical information correctly (combined risks of surgery and treatment)
If you are told "the surgery has a 90% success rate, and there is a 95% chance of no complications afterward", the chance that both go well is \(0.9 \times 0.95 = 0.855\) (85.5%). (This is a guide that treats the two as independent.)
For critical life decisions such as choosing a treatment, being able to calculate the overall outlook when several probabilities combine helps you make a decision you are comfortable with.
Quality control (the probability that all parts are good)
In a product that uses two parts, each good 99% of the time, the probability that both parts are good is \(0.99 \times 0.99 = 0.9801\) (98.01%).
Quality control staff in manufacturing repeat this calculation for every part to estimate the pass rate of the whole product, and work backward to find the defect rate allowed for each part.
Formulas and figures
Complement (probability that A does not happen)
Figure
Standard notation
(the usual math form)
\(P(A')\)
\(=\)
\(1\)
\(-\)
\(P(A)\)
In words
(symbols replaced with words)
③
\(P(A')\): probability that A does not happen
\(=\)
①
\(1\): total probability
\(-\)
②
\(P(A)\): probability that A happens
The formula in words
①
Take the
\(1\): total probability
②
subtract the
\(P(A)\): probability that A happens
③
and you get the
\(P(A')\): probability that A does not happen
Quick example
If the chance of winning a raffle is 0.2, the chance of not winning is
\(P(A')\): not winning
\(=\)
\(1\): total probability
\(-\)
winning (0.2)
\(1 - 0.2 = 0.8\ \ (80\%)\)
Key idea
The probability that something happens and the probability that it does not always add up to the total probability, 1. So subtracting the probability that it happens from 1 gives the probability that it does not. The symbol \(A'\) (read "A prime") stands for the event "A does not happen", called the complement of A. Some books write it as \(A^c\) or \(\overline{A}\) instead.
Intersection (probability that both A and B happen)
Figure
Standard notation
(the usual math form)
\(P(A \cap B)\)
\(=\)
\(P(A)\)
\(\times\)
\(P(B)\)
In words
(symbols replaced with words)
③
\(P(A \cap B)\): probability that both A and B happen
\(=\)
①
\(P(A)\): probability that A happens
\(\times\)
②
\(P(B)\): probability that B happens
The formula in words
①
Take the
\(P(A)\): probability that A happens
②
multiply it by the
\(P(B)\): probability that B happens
③
and you get the
\(P(A \cap B)\): probability that both A and B happen
Quick example
If the chance of sun is 0.5 on day 1 of a trip and 0.3 on day 2, the chance of sun on both days is
\(P(A \cap B)\): sun on both days
\(=\)
sun on day 1 (0.5)
\(\times\)
sun on day 2 (0.3)
\(0.5 \times 0.3 = 0.15\ \ (15\%)\)
Key idea
Why is "and" a multiplication? A happens in a share \(P(A)\) of all cases, and within that share, B also happens in a share \(P(B)\). It is "a share of a share", so you multiply. 30% of 50% is \(0.5 \times 0.3 = 0.15\) (15%).
The square on the right of the figure above treats the whole as a square with area 1, with width \(P(A)\) and height \(P(B)\). The case "both A and B happen" is the rectangle in the lower left, and its area is \(P(A) \times P(B)\). The Venn diagram on the left only shows which part is which; the areas of its circles do not match the sizes of the probabilities.
This formula works only when A and B do not affect each other (they are independent). It does not work when one result changes the probability of the other, as in "drawing two tickets without putting the first one back".
Union (probability that at least one of A and B happens)
Figure
Standard notation
(the usual math form)
\(P(A \cup B)\)
\(=\)
\(P(A)\)
\(+\)
\(P(B)\)
\(-\)
\(P(A \cap B)\)
In words
(symbols replaced with words)
④
\(P(A \cup B)\): probability that at least one happens
\(=\)
①
\(P(A)\): probability that A happens
\(+\)
②
\(P(B)\): probability that B happens
\(-\)
③
\(P(A \cap B)\): probability that both happen
The formula in words
①
Take the
\(P(A)\): probability that A happens
②
add the
\(P(B)\): probability that B happens
③
subtract the
\(P(A \cap B)\): probability that both happen
once
④
and you get the
\(P(A \cup B)\): probability that at least one happens
Quick example
If the chance of sun is 0.5 on day 1 and 0.3 on day 2 (so the chance of sun on both days is 0.5 × 0.3 = 0.15), the chance of sun on at least one of the days is
\(P(A \cup B)\): sun on at least one day
\(=\)
sun on day 1 (0.5)
\(+\)
sun on day 2 (0.3)
\(-\)
sun on both days (0.15)
\(0.5 + 0.3 - 0.15 = 0.65\ \ (65\%)\)
Key idea
Why subtract \(P(A \cap B)\)? If you simply add \(P(A) + P(B)\), the case "both happen" is counted twice, once as part of A and once as part of B. Subtracting the overlap \(P(A \cap B)\) once fixes the double count. This counting rule is called the inclusion-exclusion principle.
Symmetric difference (probability that exactly one happens)
Figure
Standard notation
(the usual math form)
\(P(A \Delta B)\)
\(=\)
\(P(A)\)
\(+\)
\(P(B)\)
\(-\)
\(2P(A \cap B)\)
In words
(symbols replaced with words)
④
\(P(A \Delta B)\): probability that exactly one happens
\(=\)
①
\(P(A)\): probability that A happens
\(+\)
②
\(P(B)\): probability that B happens
\(-\)
③
\(2P(A \cap B)\): twice the probability that both happen
The formula in words
①
Take the
\(P(A)\): probability that A happens
②
add the
\(P(B)\): probability that B happens
③
subtract
\(2P(A \cap B)\): twice the probability that both happen
④
and you get the
\(P(A \Delta B)\): probability that exactly one happens
Quick example
If event A has probability 0.5 and event B has probability 0.4 (so the probability that both happen is 0.5 × 0.4 = 0.2), the probability that exactly one happens is
\(P(A \Delta B)\): exactly one happens
\(=\)
A happens (0.5)
\(+\)
B happens (0.4)
\(-\)
both happen (0.2), counted twice
\(0.5 + 0.4 - 2 \times 0.2 = 0.5\ \ (50\%)\)
Key idea
For the union (at least one), you subtracted the overlap only once, so the case "both happen" stayed in. For the symmetric difference you want to remove the case "both happen" completely, so you subtract the double-counted overlap twice.
For two independent events, the basic rules are: "and" (both) is a multiplication, and "or" (at least one) is an addition minus the overlap.
Symbols and terms
Symbols
| \(P(A)\) | P of A | The probability that event A happens. P is the first letter of "probability". |
| \(A'\) | A prime (complement) | The event "A does not happen", called the complement of A. Also written \(A^c\) or \(\overline{A}\). It is found with \(P(A') = 1 - P(A)\). |
| \(\cap\) | cap (intersection) | The symbol for "and". \(A \cap B\) is the event "both A and B happen". In a Venn diagram it is the part where the two circles overlap. |
| \(\cup\) | cup (union) | The symbol for "or". \(A \cup B\) is the event "at least one of A and B happens". In a Venn diagram it is everything inside the two circles. |
| \(\Delta\) | delta (symmetric difference) | The symbol for "exactly one". \(A \Delta B\) is the event "exactly one of A and B happens (not both)". |
Terms
| event | Something that either happens or does not, such as "rolling a 6" or "rain tomorrow". In probability, such outcomes are called events and are given names like A and B. |
| independent | When the result of one event does not change how likely the other is. A die roll and a coin toss are independent. Drawing a second ticket without putting the first one back is not, because the first result changes the probability of the second. This calculator is only for independent events. |
| complement | The opposite event, that the event does not happen. If the probability of happening is 0.2, the probability of not happening (the probability of the complement) is \(1 - 0.2 = 0.8\). |
| intersection | Both of two events happen. The symbol is \(A \cap B\) (read "A and B"). For independent events, the probability is the product \(P(A) \times P(B)\). |
| union | At least one of two events happens. The symbol is \(A \cup B\) (read "A or B"). Its probability is the sum minus the overlap \(P(A \cap B)\), subtracted once. |
| symmetric difference | Exactly one of two events happens (not both). The symbol is \(A \Delta B\). In logic it is called "exclusive or" (XOR). |
| mutually exclusive | Two events that cannot happen at the same time. Example - on a single die roll, "an even number" and "an odd number" are mutually exclusive. Be careful - this is a different idea from "independent". |
| inclusion-exclusion principle | A counting rule for "or" probabilities. Simply adding counts the "both happen" case twice, so you subtract the overlap once. It is the basis of the union formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over the topics in this list is the fastest way forward.
| Basic probability (Grade 7) |
|
| Percents and decimals (Grade 6) |
|
| Multiplying fractions (Grade 5) |
|
| Sets and Venn diagrams (high school) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the probability that A does not happen (complement)
| Probability of A, P(A) | 0.2 |
| A does not happen, P(A') | =1-B1 |
Table to find the probability that both A and B happen
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| Both happen, P(A∩B) | =B1*B2 |
Table to find the probability that at least one of A and B happens
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| At least one happens, P(A∪B) | =B1+B2-B1*B2 |
Table to find the probability that exactly one happens (symmetric difference)
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| Exactly one happens, P(AΔB) | =B1+B2-2*B1*B2 |
After pasting, column A holds the labels and column B holds the numbers. The upper rows are your inputs, and the formula in the last row calculates the result from them.
In a formula, "B1" and "B2" mean "use the number in that cell", and "*" is multiplication. Just replace the input numbers with your own probabilities.
In the second table, for example, B3 shows 0.5 × 0.3 = 0.15. Only the first table (the complement) has a single input in B1, with the result in B2.
In a formula, "B1" and "B2" mean "use the number in that cell", and "*" is multiplication. Just replace the input numbers with your own probabilities.
In the second table, for example, B3 shows 0.5 × 0.3 = 0.15. Only the first table (the complement) has a single input in B1, with the result in B2.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the probability that A does not happen (complement)
| Probability of A, P(A) | 0.2 |
| A does not happen, P(A') | =1-B1 |
Table to find the probability that both A and B happen
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| Both happen, P(A∩B) | =B1*B2 |
Table to find the probability that at least one of A and B happens
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| At least one happens, P(A∪B) | =B1+B2-B1*B2 |
Table to find the probability that exactly one happens (symmetric difference)
| Probability of A, P(A) | 0.5 |
| Probability of B, P(B) | 0.3 |
| Exactly one happens, P(AΔB) | =B1+B2-2*B1*B2 |
The same formulas as in Excel work as is. Copy the whole table and paste it into cell A1. The upper rows are your inputs and the last row is the calculated result.
Just replace the input numbers with your own probabilities.
Just replace the input numbers with your own probabilities.
How to calculate it in Python
p_a = 0.5 # probability that event A happens
p_b = 0.3 # probability that event B happens
p_both = p_a * p_b # both happen, P(A∩B)
p_at_least_one = p_a + p_b - p_both # at least one, P(A∪B)
p_exactly_one = p_a + p_b - 2 * p_both # exactly one, P(AΔB)
p_neither = 1 - p_at_least_one # neither happens
print(f"Both happen: {p_both}")
print(f"At least one happens: {p_at_least_one}")
print(f"Exactly one happens: {p_exactly_one}")
print(f"Neither happens: {p_neither}")
Runs with the standard library only. Change the values of p_a and p_b at the top to the probabilities you want, and run it.
How to write it in LaTeX and other math languages (copy and paste)
Complement (probability that A does not happen)
P(A′) = 1 − P(A)
P(A') = 1 - P(A)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo>
<msup><mi>A</mi><mo>′</mo></msup>
<mo>)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo>
</mrow>
</math>
P(A') = 1 - P(A)
1 - pA
pNotA := 1 - pA;
p_not_a = 1 - p_a;
P(A′) = 1 - P(A)
Intersection (probability that both A and B happen)
P(A∩B) = P(A) × P(B)
P(A \cap B) = P(A) \times P(B)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>)</mo>
<mo>=</mo>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo>
<mo>×</mo>
<mi>P</mi><mo>(</mo><mi>B</mi><mo>)</mo>
</mrow>
</math>
P(A nn B) = P(A) xx P(B)
pA*pB
pAandB := pA*pB;
p_a_and_b = p_a*p_b;
P(A∩B) = P(A) × P(B)
Union (probability that at least one of A and B happens)
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A \cup B) = P(A) + P(B) - P(A \cap B)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo>)</mo>
<mo>=</mo>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo>
<mo>+</mo>
<mi>P</mi><mo>(</mo><mi>B</mi><mo>)</mo>
<mo>−</mo>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>)</mo>
</mrow>
</math>
P(A uu B) = P(A) + P(B) - P(A nn B)
pA + pB - pA*pB
pAorB := pA + pB - pA*pB;
p_a_or_b = p_a + p_b - p_a*p_b;
P(A∪B) = P(A) + P(B) - P(A∩B)
Symmetric difference (probability that exactly one happens)
P(AΔB) = P(A) + P(B) − 2P(A∩B)
P(A \Delta B) = P(A) + P(B) - 2P(A \cap B)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>Δ</mo><mi>B</mi><mo>)</mo>
<mo>=</mo>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo>
<mo>+</mo>
<mi>P</mi><mo>(</mo><mi>B</mi><mo>)</mo>
<mo>−</mo>
<mn>2</mn>
<mi>P</mi><mo>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>)</mo>
</mrow>
</math>
P(A Delta B) = P(A) + P(B) - 2 P(A nn B)
pA + pB - 2*pA*pB
pAxorB := pA + pB - 2*pA*pB;
p_a_xor_b = p_a + p_b - 2*p_a*p_b;
P(AΔB) = P(A) + P(B) - 2P(A∩B)
How to have ChatGPT do the calculation
You are a probability calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Event A has probability 0.5 and event B has probability 0.3. A and B are independent. Find each of the following: 1. The probability that both A and B happen, P(A∩B) 2. The probability that A or B (at least one) happens, P(A∪B) 3. The probability that exactly one of them happens, P(AΔB) 4. The probability that neither happens Show the formulas you used and the numbers from the execution result in a table.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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