A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained, the probability that 5 comes before 7 is:

0.2 0.3 0.4 0.5

0.4

P(5 comes before 7)  = P(5) + P(neighter 5 nor 7) P(5) + P(neither 5 nor 7) P(neither 5 nor 7) × P(5) + ....... $=\frac{4}{36}(1+\frac{26}{36}+\frac{26^2}{36^2}+....)$ [sum of infinite G.P. series] $=\frac{\frac{4}{36}}{1-\frac{26}{36}}=\frac{4}{10}=\frac{2}{5}$