Probability dice question, need to know if i am right? [closed]

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Suppose that we have 2 dice, one Red and one Black. The Red die is defective, the probability to get 6 is 1/3. Now if we throw the 2 dice and get equal values the Black one wins, obviously if one is bigger that one wins. Which one of the dice has the biggest probability to win?

My solution:

P(red die for not rolling six) = 1 - 1/3 = 2/3

Since there are 5 other possible values we have P(red die for each value) = 2/3 * 1/5 = 2/15 Now for the Red Die to win we have 15 possibilities: (1,2) (1,3) (1,4) (1,5) (1,6) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (4,5) (4,6) (5,6*) => Of which 5 have 6 in it. While for the Black one 21 of which 1 have 6 in it.

P(red to win) = 10 * (1/6 * 2/15) + 5 * (1/6 * 1/3) = .5 P(black to win) = 20 * (1/6 * 2/15) + 1 * (1/6 * 1/3) = .5

Both have equal possibilities to win, however from some teacher notes it says that the Black one has the biggest probability to win and i don't know why.

Please help!