2 3 If A and B are independent events show the following a A

2. 3. If A and B are independent events, show the following (a) A and Be are independent [5 marks] (b) Ac and Bc are independent |5 marks]

Solution

Ans:

This can be proven by the following theorems: P(B) + P(Bc ) = 1 and

P(A Bc ) + P(A B) = P(A)

Thus, P(A) · P(B) = P(A)[1 P(Bc )] = P(A) P(A) · P(Bc ).

Also, P(A B) = P(A) P(A Bc ).

On Substituting these into the equation P(A B) = P(A) · P(B), we have P(A) P(A B c ) = P(A) P(A) · P(B c ).

Subtracting P(A) from both sides of the equation and dividing both sides by 1, we obtain P(A B c ) = P(A) · P(B c )

Hence proved.

From above we had that P(A Bc ) = P(A) · P(Bc ). Analogously we have for P(A) and P(Ac ) that:

P(A) + P(Ac ) = 1 and P(A Bc ) + P(Ac Bc ) = P(Bc ).

Therefore, we have P(A)·P(Bc ) = [1P(Ac )]P(Bc ) = P(Bc )P(Ac )·P(Bc )

and also that P(ABc ) = P(Bc )P(Ac Bc ).

On Substituting into the equation P(ABc ) = P(A)·P(Bc ), we obtain P(B c ) P(A c B c ) =

P(B c ) P(A c ) · P(B c ). Subtracting P(Bc ) from both sides and dividing by 1, we obtain P(A c B c ) = P(A c ) · P(B c ). Therefore, Ac and Bc are independent events.