Please forgive me if this is an easy question, because I have never had a class in probability.

Lets say two people are each going to roll a D6 to determine who is going to go first in a board game. I understand that after the first person rolls (Person A), the probability that the second roller (Person B) will roll a higher number changes.

But what I don't understand is this. Before anyone rolls, before you know which person will roll first (assume they are in some kind of dice duel), how is their probability of winning the dice roll not the same? Each person seems to have an equal chance of winning the dice roll.

This came up playing games last night, and two people said that it was a illogical question because someone has to go first, i.e. one of dice will always fall first. But in my mind, you still don't know which person will roll first, so one person cannot have a different probability than the other.

So essentially I am just confused about what the probability of Person A is versus B, before you know which person rolls first. Maybe someone can explain to me why this is an illogical question, because my friends had trouble putting it into words.

A secondary question is, do you have better odds if you roll first or second?


It is conditional on what Player A rolls.

Before, or a priori, the first roll both players have the same probability of winning (assuming ties just go again).

Say the first person rolls a 1... then the second person has a probability of winning of $5/6$ and a probability of a tie of $1/6$.

Say the first person rolls a 2... then the second person has a probability of winning of $4/6$, tying $1/6$ and losing $1/6$.

And so on.

You can see that the probability of a tie is $1/6$. The probability of Player A losing with a 1 ($5/6$) is balanced by the probability of him winning with a six ($5/6$) and so on so it is fair.


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