# Independent Events And Probability

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

A bag contains (2n + 1) coins.  It is known that n of these coins have a head on both sides, whereas the remaining (n+1) coins are fair.  A coin is picked up at random from the bag and tossed.  If the probability that the  toss results in a head is 3142, then n is equal to: