Two trains P and Q are scheduled to reach New Delhi railway station at 10.00 AM. The probability that train P and train Q will be late is 7/9 and 11/27 respectively. The probability that train Q will be late, given that train P is late, is 8/9. Then the probability that neither train will be late on a particular day is

Let 'A' and 'B' be the event of train reaching at the station respectively. 

P(A)$$_{\text{Late}}$$ = $$\dfrac{7}{9}$$, therefore, P(A)$$_{\text{On time}}$$ = $$\dfrac{2}{9}$$.

P(B)$$_{\text{Late}}$$ = $$\dfrac{11}{27}$$, therefore, P(B)$$_{\text{On time}}$$ = $$\dfrac{16}{27}$$.

The probability that train Q will be late, given that train P is late, is 8/9.

P$$(\dfrac{B_{\text{Late}}}{A_{\text{Late}}})$$=$$\dfrac{8}{9}$$

P(A$$_{\text{Late}} \cap$$ B$$_{\text{Late}})$$ = P(A)$$_{\text{Late}}$$*P($$\dfrac{B_{\text{Late}}}{A_{\text{Late}}})$$

P(A$$_{\text{Late}} \cap$$ B$$_{\text{Late}})$$ = $$\dfrac{7}{9}$$*$$\dfrac{8}{9} = \dfrac{56}{81}$$

Therefore, the probability that neither train is late = 1 - (P(A)$$_{\text{Late}}$$+P(B)$$_{\text{Late}}$$ - P(A$$_{\text{Late}} \cap$$ B$$_{\text{Late}})$$)

$$\Rightarrow$$ 1 - ($$\dfrac{7}{9}$$+$$\dfrac{11}{27}$$-$$\dfrac{56}{81}$$)

$$\Rightarrow$$ $$\dfrac{41}{81}$$

Hence, we can say that option B is the correct answer.