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Independent Events and Non-Exclusive Events

Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined.

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Student-friendly explanation

Independence is about no change in probability under given information: P(A|B)=P(A) and P(B|A)=P(B). It is different from mutually exclusive events. Mutually exclusive events cannot occur together, while independent events can occur together unless one event has probability zero.

How to write this in exams

  1. 1

    Start with the exact idea

    Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined.

  2. 2

    Then show how to use it

    Write P(A), P(B), and P(A∩B). If P(A∩B) is not directly given, find it using the union formula. Compute P(A)P(B). Compare exactly, preferably as fractions. State the conclusion clearly.

  3. 3

    Add one concrete example

    When a fair coin is tossed and a fair die is rolled, let A be getting a head and B be getting an even number. P(A)=1/2, P(B)=1/2, and P(A∩B)=1/4, so A and B are independent.

  4. 4

    Avoid this incomplete answer

    Saying events are independent because they are not mutually exclusive, without checking P(A∩B)=P(A)P(B).

Definition

Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined.

Example

When a fair coin is tossed and a fair die is rolled, let A be getting a head and B be getting an even number. P(A)=1/2, P(B)=1/2, and P(A∩B)=1/4, so A and B are independent.

Rule to remember

Independence test: A and B independent iff P(A∩B)=P(A)P(B). Equivalent forms: P(A|B)=P(A) when P(B)>0, and P(B|A)=P(B) when P(A)>0.

Memory hook

Independent means probability unchanged, not events separated.

Examples and method

Worked example

Given P(A)=2/5, P(B)=3/4, and P(A∪B)=17/20, test independence. First find P(A∩B)=P(A)+P(B)-P(A∪B)=2/5+3/4-17/20=8/20+15/20-17/20=6/20=3/10. Now P(A)P(B)=(2/5)(3/4)=6/20=3/10. Since both values are equal, A and B are independent.

Method to apply

Write P(A), P(B), and P(A∩B). If P(A∩B) is not directly given, find it using the union formula. Compute P(A)P(B). Compare exactly, preferably as fractions. State the conclusion clearly.

Diagram support

A Venn diagram may show that independent events can overlap. It should not be used as the only proof; numerical equality is required.

How CBSE asks it

Questions may ask to prove independence, find a missing probability using independence, or distinguish independent events from mutually exclusive events in assertion-reason format.

Avoid common mistakes

Common confusion

Students often think mutually exclusive events are independent. If A and B are mutually exclusive with positive probabilities, then P(A∩B)=0 but P(A)P(B)>0, so they are not independent.

Common wrong answer

Saying events are independent because they are not mutually exclusive, without checking P(A∩B)=P(A)P(B).

Exam tip

To test independence, compare P(A∩B) with P(A)P(B). Do not decide from wording alone.

Quick check

If P(A)=0.3, P(B)=0.5, and P(A∩B)=0.15, are A and B independent?

Yes. P(A)P(B)=0.3×0.5=0.15=P(A∩B), so A and B are independent.

Answer writing and exam use

1-mark answer

Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined.

2-mark answer

Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined. Independence test: A and B independent iff P(A∩B)=P(A)P(B). Equivalent forms: P(A|B)=P(A) when P(B)>0, and P(B|A)=P(B) when P(A)>0. When a fair coin is tossed and a fair die is rolled, let A be getting a head and B be getting an even number. P(A)=1/2, P(B)=1/2, and P(A∩B)=1/4, so A and B are independent.

3-mark answer

Independence is about no change in probability under given information: P(A|B)=P(A) and P(B|A)=P(B). It is different from mutually exclusive events. Mutually exclusive events cannot occur together, while independent events can occur together unless one event has probability zero. Independence test: A and B independent iff P(A∩B)=P(A)P(B). Equivalent forms: P(A|B)=P(A) when P(B)>0, and P(B|A)=P(B) when P(A)>0. Given P(A)=2/5, P(B)=3/4, and P(A∪B)=17/20, test independence. First find P(A∩B)=P(A)+P(B)-P(A∪B)=2/5+3/4-17/20=8/20+15/20-17/20=6/20=3/10. Now P(A)P(B)=(2/5)(3/4)=6/20=3/10. Since both values are equal, A and B are independent. Questions may ask to prove independence, find a missing probability using independence, or distinguish independent events from mutually exclusive events in assertion-reason format. Saying events are independent because they are not mutually exclusive, without checking P(A∩B)=P(A)P(B).
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