Independent Events: Formula, Rules and Examples
Independent Events are events where the occurrence of one event does not change the probability of the other. This page explains the multiplication rule, conditional probability test, complement method, difference between independent and mutually exclusive events, and solved probability examples for aptitude questions.
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What Are Independent Events?
For independent events, P(B | A) = P(B) and P(A | B) = P(A), provided the conditioning events have positive probability. For example, tossing a coin and rolling a die are independent because the coin result does not change the die result. The probability of getting a head and an even number is 1/2 × 3/6 = 1/4.
Independent Events Video Lessons
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Independent Events and Multiplication Theorem
Learn how to identify independent events and apply the multiplication theorem to calculate the probability of their joint occurrence.
Independent Events Formula & Tricks
Important Formulas
Use this when A and B are independent and the question asks for both events to occur.
These equalities indicate that the occurrence of one event does not change the probability of the other.
The complements of independent events are also independent.
For independent events, substitute P(A ∩ B) = P(A)P(B) into the addition rule.
Quick Tricks
For several independent events, calculate the probability that none occurs and subtract it from 1.
Do not multiply probabilities directly unless the events are stated to be independent or the conditional probability confirms independence.
For n independent trials with success probability p, the probability of exactly r successes is C(n,r)pʳ(1 − p)ⁿ⁻ʳ.
Independent Events Concepts
Multiplication Rule for Independent Events
If P(A) = a and P(B) = b, then P(A ∩ B) = ab. For three mutually independent events, P(A ∩ B ∩ C) = P(A)P(B)P(C).
Testing Whether Events Are Independent
Calculate the intersection probability and compare it with the product of the individual probabilities. If the values are equal, the events are independent; otherwise, they are dependent. Equivalently, test whether P(A | B) = P(A), when P(B) is non-zero.
Independent Events and Complements
The complement rule gives P(Aᶜ) = 1 − P(A). Thus, for independent A and B, P(Aᶜ ∩ B) = [1 − P(A)]P(B), and P(Aᶜ ∩ Bᶜ) = [1 − P(A)][1 − P(B)].
Independent Versus Mutually Exclusive Events
For mutually exclusive events, P(A ∩ B) = 0. If both events have positive probabilities, they cannot also be independent because P(A)P(B) is positive. Therefore, non-zero mutually exclusive events are dependent.
Independent Events Data Table
Use the structured data below while following the explanation and examples.
Independent and Mutually Exclusive Events
Comparison of the intersection rule and occurrence relationship.
| Property | Independent events | Mutually exclusive events |
|---|---|---|
| Can both occur? | Yes, generally | No |
| Intersection probability | P(A ∩ B) = P(A)P(B) | P(A ∩ B) = 0 |
| Effect of one event | Does not change the other probability | Occurrence of one rules out the other |
| Can both conditions hold? | Only if at least one event has probability 0 | Not when both probabilities are positive |
Note: Mutually exclusive events with positive probabilities are dependent.
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Independent Events Revision Points
Recall the following definitions, formulas and checks.
- Independent events satisfy P(A ∩ B) = P(A)P(B).
- Equivalent tests are P(A | B) = P(A) and P(B | A) = P(B), when defined.
- For independent events, P(A ∪ B) = P(A) + P(B) − P(A)P(B).
- Complements of independent events are also independent.
- For n independent trials, exactly r successes have probability C(n,r)pʳ(1 − p)ⁿ⁻ʳ.
- Mutually exclusive events with positive probabilities are not independent.
Independent Events FAQs
What is the formula for two independent events?
The formula is P(A ∩ B) = P(A)P(B), where A ∩ B means that both events occur.
If P(A) = 0.4 and P(B) = 0.5, what is P(A and B) when the events are independent?
P(A ∩ B) = 0.4 × 0.5 = 0.20.
How can independence be checked from given probabilities?
Compare P(A ∩ B) with P(A)P(B). The events are independent only when these two values are equal.
Are mutually exclusive events always independent?
No. If mutually exclusive events both have positive probabilities, P(A ∩ B) = 0 but P(A)P(B) > 0, so they are dependent.
What is the probability of at least one success in three independent trials?
If each trial has success probability p, the probability is 1 − (1 − p)³.
What is the probability of exactly two successes in four independent trials?
If each trial has success probability p, the probability is C(4,2)p²(1 − p)² = 6p²(1 − p)².