HTH, THH} and C = {at most two heads} = {HHT, HTH, HTT, THH, TTH, THT, TTT} Also (A ∩ B) = {HHH}; (A ∩ C) = {TTT} and (B ∩ C) ={HHT, HTH, THH} ∴ P(A) = ; P ( B ) = ; P(C) = and P(A ∩ B) = , P (A ∩ C) = , P(B ∩ C) = Also P(A). P(B) = ⋅ P(A). P(C) = ⋅ and P(B). P(C) = ⋅ Thus, P(A ∩ B) = P(A).
P(B) P(A ∩ C) ≠ P(A) . P(C) and P(B ∩ C) ≠ P(B). P(C) Hence, the events ( A and B ) are independent, and the events ( A and C ) and ( B and C ) are dependent. - - Example .
A can solve per cent of the problems given in a book and B can solve per cent. What is the probability that at least one of them will solve a problem selected at random? Given the probability that A will be able to solve the problem = and the probability that B will be able to solve the problem = i.e., P(A) = and P(B) = P A ( ) = –P(A) = P B ( ) = – P(B) = P (at least one solve the problem) = P(A B) = − ( P A = − P A = − ( ) ⋅ ( ) P A P B = Hence the probability that at least one of them will solve the problem = . Example .
A bag contains white and black balls. Two balls are drawn at random one after the other without replacement. Find the probability that both balls drawn are black. Let A , B be the events of getting a black ball in the first and second draw.
Probability of