Choose the probability rule that matches the events
Enter known probabilities, choose the event relationship, and see which rule is valid before the numbers are combined.
Result
There is a 0.58 probability that at least one of the two independent events occurs.
P(A∪B)=0.58
Conditions
Every probability must lie from 0 to 1 inclusive.
Multiplying P(A) by P(B) is valid here only because independence is stated.
The addition rule subtracts the intersection so outcomes in both events are not counted twice.
Conditional probability P(A given B) requires P(B) greater than zero.
Steps
Translate the wording The phrase at least one means the union of A and B.P(A or B)=P(A∪B)
Find the independent intersection Independence makes the conditional probability of A given B equal to P(A).P(A∩B)=P(A)P(B)=(0.4)(0.3)=0.12
Apply the addition rule Add the event probabilities and subtract their overlap once.P(A∪B)=0.4+0.3−0.12=0.58
Check the complement Neither event occurs with probability 0.42, so at least one occurs with probability 0.58.1−(1−0.4)(1−0.3)=1−(0.6)(0.7)=0.58
✓
Independent check
The direct union calculation and the independent complement calculation both give 0.58, and the result lies between the larger individual probability, 0.4, and 1.
Scope
What this independent-event-union covers
If P(A) = 0.4 and P(B) = 0.3 for independent events, then P(A and B) = 0.12 and P(A or B) = 0.58. Independence permits multiplication for the intersection, but the union still subtracts that overlap.
Complements
Find the probability that an event does not occur from one known event probability.
Examples: P(A)=0.72, P(A^c)=1-0.72=0.28
Unions with known overlap
Add two event probabilities and subtract a supplied intersection.
Examples: P(A)=0.5,\ P(B)=0.4,\ P(A\cap B)=0.2
Intersections
Use a stated conditional probability for dependent events or multiply marginal probabilities when independence is given.
Examples: P(A\cap B)=P(A\mid B)P(B), P(A\cap B)=P(A)P(B)\text{ if independent}
Conditional probability
Restrict the sample space to the conditioning event and divide the intersection by its probability.
Examples: P(A\mid B)=0.18/0.30=0.60
How to use it
Enter enough information for one clear task
1
Define the events
Write what A and B mean before entering numbers, including whether the question asks for and, or, not, or given.
2
State the relationship
Select independent, mutually exclusive, or neither only when the problem provides enough information to justify it.
3
Enter known probabilities
Provide the marginals, intersection, or conditional probability required by the selected rule.
4
Check the result range
A valid probability is between 0 and 1 and must be consistent with the events it contains.
Worked inputs
Examples to try
Use these examples to recognize the method, compare equivalent forms, and check your own work.
Complement of an event
Subtract the event probability from 1.
P(A)=0.72
Expected result
P(Ac)=1−0.72=0.28
Union with overlap
Apply the general addition rule.
P(A)=0.5,P(B)=0.4,P(A∩B)=0.2
Expected result
P(A∪B)=0.5+0.4−0.2=0.7
Independent intersection
Multiply because independence is stated.
P(A)=0.6,P(B)=0.5
Expected result
P(A∩B)=(0.6)(0.5)=0.3
Conditional probability
Divide the intersection by the probability of the conditioning event.
P(A∩B)=0.18,P(B)=0.30
Expected result
P(A∣B)=0.18/0.30=0.60
Exactly one of two events
Remove the intersection from each event, so subtract it twice from the marginal sum.
P(A)=0.40,P(B)=0.35,P(A∩B)=0.10
Expected result
P(exactly one)=0.40+0.35−2(0.10)=0.55
Equally likely outcomes
Divide the favorable count by the total count when all outcomes are equally likely.
3 favorable outcomes out of 10
Expected result
P(E)=3/10=0.3
Complete example
Find the probability that at least one independent event occurs
At least one means A or B. The events can overlap, so the overlap must be removed from the sum even though they are independent.
P(A)=0.4,P(B)=0.3
1
Calculate the overlap
Independence allows the multiplication rule to use the two marginal probabilities.
P(A∩B)=(0.4)(0.3)=0.12
2
Calculate the union
Subtract the 0.12 overlap after adding 0.4 and 0.3.
P(A∪B)=0.4+0.3−0.12=0.58
3
Verify through neither
For independent events, the complements are also independent.
P(Ac∩Bc)=(0.6)(0.7)=0.42
4
Take the complement
At least one and neither are complementary outcomes.
1−0.42=0.58
P(A∪B)=0.58
Verification: Both the addition rule and the complement of neither give 0.58. The answer is at least 0.4 because the union contains A, and it is no more than 0.7 because overlap reduces the simple sum.
Avoidable errors
Common mistakes and how to fix them
Adding without subtracting overlap
Problem: P(A\cup B)=P(A)+P(B)
Why it matters: This counts outcomes in both A and B twice unless the events are mutually exclusive.
Better approach: Use the general addition rule and subtract P(A intersection B).
Multiplying without independence
Problem: P(A\cap B)=P(A)P(B)\text{ for every pair of events}
Why it matters: Dependence can change the chance of A once B is known.
Better approach: Use P(A given B) times P(B), or multiply marginals only when independence is established.
Confusing independent with mutually exclusive
Problem: Assuming events that cannot occur together are independent.
Why it matters: For nonzero mutually exclusive events, occurrence of one makes the other impossible, so the events are dependent.
Better approach: Use intersection zero for mutually exclusive events; use unchanged conditional probability for independent events.
Ignoring the conditioning event
Problem: P(A\mid B)=P(A\cap B)
Why it matters: Conditional probability measures the intersection relative to the reduced sample space B.
Better approach: Divide by P(B), and confirm P(B) is greater than zero.
Trust the result for the right reasons
Checks, assumptions, and limits
How results are checked
Every input and result is checked to lie from 0 to 1 inclusive.
Union results are checked against the bounds max(P(A), P(B)) and min(1, P(A) + P(B)).
Conditional results are multiplied back by the conditioning probability to recover the stated intersection.
Independent at-least-one results are recomputed as one minus the probability that neither event occurs.
When to stop and revise the input
The event relationship must come from the problem context; a calculator cannot infer independence from two marginal probabilities.
Conditional probability is undefined when the conditioning event has probability zero.
Equally likely counting applies only when the outcomes truly have equal probability.
The page handles basic event rules, not a complete probability distribution model or statistical inference.
Common questions
Choose the probability rule that matches the events FAQ
When do I add probabilities?
Add when the question asks for A or B, then subtract P(A and B) to remove double counting. If the events are mutually exclusive, their intersection is zero.
When do I multiply probabilities?
For an intersection, use P(A given B) times P(B). If A and B are independent, P(A given B) equals P(A), so the rule simplifies to P(A) times P(B).
Are independent events mutually exclusive?
Usually not. Independent events do not change each other's probabilities and can occur together. Two nonzero mutually exclusive events cannot occur together and therefore are not independent.
What does P(A given B) mean?
It is the probability of A after restricting attention to outcomes where B occurred. Algebraically it is P(A and B) divided by P(B), provided P(B) is greater than zero.
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