$A$ and $B$ play a game with colored balls. $A$ starts with $6$ balls: $2$ orange, $2$ yellow and $2$ green. $B$ starts with $4$ balls: $2$ pinks and $2$ gray. Player "$A$" plays first, and both alternate turns. On each turn, the player places a ball at random on the table. The ball is placed there for rest of the game. Player wins and game ends when they have placed two balls of the same color on the table. Find the probability that $A$ wins the game.
My son tried forming a spiral case and got stuck at the conditional probability. In this case can we assume the total probability to be $1$. I am asking this because here cases are not the same $B$ has a clear advantage because of less balls and thus can win faster.
Kindly provide help for the above problem. (Please use basic probability terms and methods.) Suggest additional tags as well. Thank you.