In a casino, the gambling machine determines whether or not a gambler will win depending on the past chances of winning through the previous two games.

If the gambler lost the first and second game, then the probability of losing the third game is 0.6

If the gambler lost the first and won the second game, then the probability of losing the third game is 0.1

If the gambler won the first game and lost the second game, then the probability of losing the third game is 0.2

If the gambler won the first and second game, then the probability of losing the third game is 0.5

Find the proportion of games in where the gambler loses

(Hint: Use a 4 by 4 matrix)