I'll start with the hint I gave yesterday,

Hint: if the first card drawn isn't a joker, then the probability player 2 wins is the same as probability player 1 wins in a game with k-1 non-jokers.

You can use that to break into 2 cases and combine with them with conditional probability, and get a recursive formula.

P_k = 1/(k+1) + (k/(k+1))(1 - P_(k-1))

The 1/(k+1) is the probability of picking the joker on the first draw, which wins for player 1, and if not -- probability k/(k+1) -- then player 1 wins if player 2 loses.

Past that it's algebra to show that the given formula for odd P_k comes from the formula for even P_(k-1) and same swapping odd and even.