Try to use DP to solve this problem

Let $$$dp[i][p][q][turn]$$$ denote the probability of Rajat winning the game, where:

• $$$i$$$ denotes the number of turns left,
• $$$p$$$ denotes the parity of Rajat’s score,
• $$$q$$$ denotes the parity of Dr. S’s score, and
• $$$turn$$$ is 0 if it is Rajat’s turn and 1 if it is Dr. S’s turn.

Let $$$p_{\text{even}}$$$ and $$$p_{\text{odd}}$$$ denote the probabilities of landing even and odd elements, respectively, among the first $$$n - 1$$$ elements of $$$a$$$. We will consider the scenario where $$$a_n$$$ is rolled on the die separately.

The transitions will be as follows:

If it is Rajat’s turn, then there are 2 additional cases:

Case 1: When $$$a_n$$$ is odd

Case 2: When $$$a_n$$$ is even

Similarly, for Dr. S’s turn:

Case 1: When $$$a_n$$$ is odd

Case 2: When $$$a_n$$$ is even

where $$$p_n$$$ is the probability of rolling $$$n$$$.