For the three events A,B and C,$$P($$ exactly one of the events A or B occurs $$)=P($$ exactly one of the two events B or C occurs $$)=P($$ exactly one of the events C or A occurs $$)=p$$ and $$P($$ all the three events occur simultaneously $$)=p2$$ where $$0$$

Question

For the three events A,B and C,$$P($$ exactly one of the  events A or B occurs $$)=P($$ exactly one of the two events B or C occurs $$)=P($$ exactly one of the events C or A occurs $$)=p$$ and $$P($$ all the three events occur simultaneously $$)=p2$$ where $$0$$

Infinity Learn · Accepted Answer

We know $$thatP(exactly$$ one of A or B $$occurs)$$ $$=P(A)+P(B)$$−$$2P(A$$∩$$B)Therefore$$,$$P(A)+P(B)$$−$$2P(A$$∩$$B)=p------1Similarly$$,$$P(B)+P(C)$$−$$2P(B$$∩$$A)=p----2andP(C)+P(A)$$−$$2P(C$$∩$$A)=p-----3Adding$$ Eqs. (1), (2) and (3) we $$get2$$[$$P(A)+P(B)+P(C)$$−$$P(A$$∩$$B)$$−$$P(B$$∩$$C)$$−$$P(C$$∩$$A)$$]$$=3p$$⇒ $$P(A)+P(B)+P(C)$$−$$P(A$$∩$$B)$$−$$P(B$$∩$$C)$$−$$P(C$$∩$$A)=3p/2---4It$$ is also given that $$P(A$$∩B∩$$C)=p2----5Now$$,$$P(at$$ least one of A, B and $$Q)=P(A)+P(B)+P(C)$$−$$P(A$$∩$$B)$$−$$P(B$$∩$$C)$$−$$P(C$$∩$$A)+P(A$$∩B∩$$C)=3p2+p2=3p+2p22$$          [Using Eqs. (4) and (5)]

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material