A and B ) = P ( A ∩ B ) = P ( A ) P ( B ) The theorem can be extended to three or more independent events. Thus for three events the theorem states that P ( A and B and C ) = P A P A P B P C = ( ) ( ) ( ) Example . An unbiased die is thrown. If A is the event ‘the number appearing is a multiple of ’ and B be the event ‘the number appearing is even’ number then find whether A and B are independent?
We know that the sample space is S = { , , , , , } Now, A = { , } ; B = { , , } then (A ∩ B) ={ } P ( A ) = = P ( B ) = = and P ( A ∩ B ) = Clearly P ( A ∩ B ) = P ( A ) P ( B ) Hence A and B are independent events. Example . Let P ( A ) = and P ( B ) = . Find P ( A ∩ B ) if A and B are independent events.
Since A and B are independent events then P ( A ∩ B ) = P ( A ) P ( B ) Given that P ( A ) = and P(B) = , then P ( A ∩ B ) = × = Example . Three coins are tossed simultaneously. Consider the events A ‘three heads or three tails’, B ‘atleast two heads’ and C ‘at most two heads’ of the pairs ( A,B ), ( A,C ) and ( B,C ), which are independent? Which are dependent?
Here the sample space of the experiment is S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT} A = {Three heads or Three tails} = {HHH, TTT} B = {at least two heads} = {HHH, HHT,