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What is the probability that one die will win against another, meaning when you roll them you get a larger number if the numbers on their faces are: A: 1,4,4,4,4,4 B: 2,2,2,5,5,5

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You need to translate the event $A>B$ into the actual die rolls (note $A=B$ is impossible):

  • A rolls a 1: any B will be bad
  • A rolls a 4: B will be good when rolls a 5, but bad when rolls a 2.

What is the probility of each case and its subcases?

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Let us find the probability $A$ beats $B$. Clearly, $A$ must get a $4$ (probability $\frac{5}{6}$) and $B$ must get a $2$ (probability $\frac{3}{6}$). Since the results of the throws are independent, the probability $A$ gets a $4$ and $B$ gets a $2$ is $\frac{5}{6}\cdot\frac{3}{6}$.

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