I’m struggling to understand how a thrown dart could land on an irrational number. It seems definitionally that any physically realized outcome must pertain to a rational number because it is impossible to physically measure one at any level of precision.
It is possible to write a random process that returns 5 or pi with 50/50 odds so this isn’t a very compelling argument that it’s possible. I don’t feel the semantics of picking a random point along a number line is gg solved just by appealing to the existence of uncountably infinite irrationals.
By most people’s definitions of random points along the number line, including the dart throw, it seems to me the probability of getting an irrational is 0.
Invoking the number of possible outcomes has bad feeling implications. For example if your set is 1 2 3 pi 4, then the probability of getting an outcome in [3,4) is higher than [2,3) and that seems like it’s breaking the intuition of what the line represents. Like as a stupid example say we only include the irrational numbers between 9 and 10 and pick a random point between 1 and 10. If the random method uniformly sampled a point along the line by distance we would suggest a 90% chance of getting a rational number <= 9 and a 10% chance of getting an irrational number above 9.
But if we sample by naive odds you’d probably claim there’s a near 100% chance of getting an irrational number above 9 because there’s an uncountable infinity up there.
That seems dumb.
It is possible to write a random process that returns 5 or pi with 50/50 odds so this isn’t a very compelling argument that it’s possible. I don’t feel the semantics of picking a random point along a number line is gg solved just by appealing to the existence of uncountably infinite irrationals.
By most people’s definitions of random points along the number line, including the dart throw, it seems to me the probability of getting an irrational is 0.
Invoking the number of possible outcomes has bad feeling implications. For example if your set is 1 2 3 pi 4, then the probability of getting an outcome in [3,4) is higher than [2,3) and that seems like it’s breaking the intuition of what the line represents. Like as a stupid example say we only include the irrational numbers between 9 and 10 and pick a random point between 1 and 10. If the random method uniformly sampled a point along the line by distance we would suggest a 90% chance of getting a rational number <= 9 and a 10% chance of getting an irrational number above 9.
But if we sample by naive odds you’d probably claim there’s a near 100% chance of getting an irrational number above 9 because there’s an uncountable infinity up there.
That seems dumb.