If Sania loses the first set, then the values of R such that her probability of winning the match is still larger than

Question

Sania Mirza is to play with Sharapova in a three set match. For a particular set, the probability of Sania winning the set is √y and if she wins probability of her winning the next set becomes y else the probability that she wins the next one becomes y². There is no possibility that a set is to be abandoned. R is the probability that Sania wins the first set.

If Sania loses the first set, then the values of R such that her probability of winning the match is still larger than that of her loosing are given by

(A) R ∈(1/2, 1)

(B) R ∈ [(1/2)^1/3, 1]

(C) R ∈ [(1/2)^3/2, 1]

(D) No values of R

Answer 1

Answer is (B) R ∈ [(1/2)^1/3, 1]

Probability that Sania wins the match = (1 – R)R² R = R³(1 – R)

Probability Sania loses the match = (1 – R)R²(1 – R) + (1 – R) (1 – R²)

Hence, 

R³ (1 – R) > (1 – R)R²(1 – R) + (1 – R) (1 – R²)

⇒ R³ > R²(1 – R) + (1 – R²) = 1 – R³