the work in the second picture is wrong. please solve differently

Wolverine has $12 and Spartan has $10. They decide to play a game. There are two boxes and each box has two balls, one blue ball and one green ball. Wolverine and Spartan chooses one box and they draw one ball without looking inside of their box at the same time. If the two balls are the same color, Wolverine gets the two dollars (that is, Wolverine earns one dollar because a player pays $1 for a game at every time.) If the two balls are different colors, Spartan gets the two dollars. They will stop playing a game until one player has all the money. What is the probability Wolverine will get all the money? 1 1 1 1 + X 2 2 2 2 1 2 Probability (p) that both the balls are of the same colors (Blue, Blue + Green, Green) = Thus Probability (a) that both the balls are of different colors (Blue, Green + Green, Blue) = 1 1 1 1 1 2 2 2 2 2 Now suppose probability that Wolverine wins with k dollars, let it is denoted by Wx, then W. = p.WH-1 +qW-1, where W. = 0 and W,2 = 1 as there are only 10 + 12 = $22. Here p = q = 1/2 =WWW- This is an homogenous difference equation, and the characteristic polynomial is or 22 - 22 +1 -0 or 2 = 1,1 Thus W. =C+ck, now putting the values W = 0 and W., =1. C1 = 0 and 9 = 22 W- 126 Here Wolverine has $ 12, hence probability that he wins = 2 1 k 22 22 11 Wolverine has $12 and Spartan has $10. They decide to play a game. There are two boxes and each box has two balls, one blue ball and one green ball. Wolverine and Spartan chooses one box and they draw one ball without looking inside of their box at the same time. If the two balls are the same color, Wolverine gets the two dollars (that is, Wolverine earns one dollar because a player pays $1 for a game at every time.) If the two balls are different colors, Spartan gets the two dollars. They will stop playing a game until one player has all the money. What is the probability Wolverine will get all the money? 1 1 1 1 + X 2 2 2 2 1 2 Probability (p) that both the balls are of the same colors (Blue, Blue + Green, Green) = Thus Probability (a) that both the balls are of different colors (Blue, Green + Green, Blue) = 1 1 1 1 1 2 2 2 2 2 Now suppose probability that Wolverine wins with k dollars, let it is denoted by Wx, then W. = p.WH-1 +qW-1, where W. = 0 and W,2 = 1 as there are only 10 + 12 = $22. Here p = q = 1/2 =WWW- This is an homogenous difference equation, and the characteristic polynomial is or 22 - 22 +1 -0 or 2 = 1,1 Thus W. =C+ck, now putting the values W = 0 and W., =1. C1 = 0 and 9 = 22 W- 126 Here Wolverine has $ 12, hence probability that he wins = 2 1 k 22 22 11