A company is migrating its database, and two software engineers, Ishaan and Kavya , take turns running a data-sync script that has a constant success rate of 3 8 per attempt. If Ishaan initiates the first attempt and they persist until the migration is successful, what is the probability that Kavya is the one who initiates the successful sync?

Step-by-step solution

Let p be the probability of success and q be the probability of failure in a single attempt. p = 3 8 , q = 1 - 3 8 = 5 8 Ishaan makes the 1st, 3rd, 5th, ... attempts. Kavya makes the 2nd, 4th, 6th, ... attempts. The probability that Kavya initiates the successful sync is the sum of probabilities that the success occurs on the 2nd, 4th, 6th, ... attempts. P( Kavya ) = qp + q^3p + q^5p + This is an infinite geometric series with first term a = qp and common ratio r = q^2 . P( Kavya ) = qp 1 - q^2 Substituting the val