My brothers and I (three of us) decided to roll a single die to see who would get to have my fathers old pocket knife. The guy who rolled the highest number would win. So I rolled first and got a 6, thus ending the contest, I got the knife.
But I'm not comfortable with this. My brothers did not even get to roll the die! My question is: Was the way I won the knife statistically fair? If not, what is the fair way to do this with a single die?
The way you did it was unfair. Why? Firstly, the aim of the game was to get the highest number, not the highest number first and this goes to the heart of the reason as to why the way you did it was unfair.
Had you said "the one to roll the highest number first wins" yes this would have been a fair win, as you had rolled a six (highest number you can roll on a normal, fair six-sided die), but you didn't say first, which means that had they both rolled a six after you had rolled your six (i.e you all roll a six), based on the rules of the game, there would have been no winners or losers. Or had one of your brothers also rolled a six and the other hadn't, still none of you could have claimed they had rolled a higher number than the other (i.e 6 is not greater than or less than six. 6 = 6).
So based by your rules, in order for this to have been a fair win you must have had your brothers each take their turn at rolling the same dice once, even after you had rolled the six, and had still rolled the highest number between the three of you. Since this was not the case, no; the way you won was not statistically fair.
How should you have done it for it to have been statistically fair? Ensure everyone gets the same number of rolls with the same exact dice. If the rules are to simply roll the highest number, count a case where at least two people both roll the same number, and that number happens to be the highest number for that round, as a tie and have only the people who rolled that number to roll again until only one person rolls the highest number in that round.