Probability of Rolling Two Dice

Question

My friend and I are playing a game and the person that rolls the highest number on the dice wins the game. Both dice has 6 sides.

Dice number 1 has the number 5 on two sides and the number 2 on four sides and is the dice used by me.

Dice number 2 has the number 4 on three sides and the number 1 on three sides and is the used by my friend.

What is the probability of me winning against my friend?

I know that I can win by

I roll 5 or my friend rolls 4

I roll 5 or my friend rolls 1

I roll 2 or my friend rolls 1

Answer 1

only way you can lose is he gets 4 an you get 2

he has a 3/6 = 1/2 chance of getting a 4

you have a 4/6 = 2/3 chance of getting 2

you have a 1/2 x 2/3 chance of losing = 1/3

if you have a 1/3 chance of losing, you have 1- 1/3 = 2/3 chance of winning

OR

you roll a 5 he rolls 4, that's 1/3 x 1/2 = 1/6

you roll a 5 he rolls 1 that's 1/3 x 1/2 = 1/6

you roll a 2, he rolls 1, that's 2/3 x 1/2 = 1/3

add them up: 1/6 + 1/6 + 1/3 = 2/3