Independent Events And Probability

Independent events

If probability of occurrence of an event A is not affected by occurrence of another event B, then A and B are said to be independent events.

Consider an example of rolling a die. If A is the event ‘the number appearing is odd’ and B be the event ‘the number appearing is a multiple of 3’, then

P(A)= \( \frac 36 \) = \( \frac 12 \) and P(B) = \( \frac 26 \)\( \frac 13 \)

Also A and B is the event ‘the number appearing is odd and a multiple of 3’ so that

P(A ∩ B) = \( \frac 16 \)

P(A│B) = \( \frac {P(A ∩ B)}{ P(B)} \)

= \( \frac {\frac 16}{\frac 13 } \)\( \frac 12 \)

P(A) = P(A│B) = \( \frac 12 \)  , which implies that the occurrence of event B has not affected the probability of occurrence of the event A .

If A and B are independent events , then P(A│B) = P(A)

Using Multiplication rule of probability, P(A ∩ B) = P(B) .P(A│B)

P(A ∩ B) = P(B) .P(A)

NOTE: A and B are two events associated with the same random experiment, then A and B are known as independent events if P(A ∩ B) = P(B) .P(A)

What are mutually exclusive events?

Two events A and B are said to be mutually exclusive events if they cannot occur at the same time. Mutually exclusive events never have an outcome in common.

Differences between independent events and mutually exclusive events:

Independent Events Mutually exclusive events
They cannot be specified based on the outcome of a maiden trial. They are independent of trials
Can have common outcomes Can never have common outcomes
If A and B are two independent events, then

P(A ∩ B) = P(B) .P(A)

If A and B are two mutually exclusive events, then

P(A ∩ B) = 0

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Practise This Question

Let ‘A’, ‘B’ and ‘C’ be three independent events with P(A)=13,P(B)=12  and P(C)=14.  The probability of exactly 2 of these events occurring, is equal to: