Randomness vs. Ignorance

A dis­tinc­tion I don’t see made of­ten enough is be­tween what I call ran­dom­ness and ig­no­rance. Roughly, ev­ery ex­pres­sion of un­cer­tainty is ei­ther about “where in the uni­verse am I?” or “what is the uni­verse like?” (or both). The former is the do­main of ran­dom­ness, the lat­ter of ig­no­rance.

Sup­pose you roll a die. You know that you’re in a situ­a­tion where you’ve just rol­led a die, and that, in roughly 1/​6th of the situ­a­tions where one has just rol­led a die, the die will come up a three. Thus, your un­cer­tainty about the die roll is ran­dom.

Sup­pose you’re won­der­ing whether or not an om­nipo­tent and im­mor­tal be­ing ex­ists. What­ever the an­swer is, it is the same ev­ery time some­one is ask­ing this ques­tion. Thus there is no ran­dom­ness in­volved, but you are ig­no­rant of what the an­swer is (though you might have a hunch).

Often, your un­cer­tainty will have com­po­nents of both. Sup­pose some­one hands you a die, you roll it five times, and ev­ery time you roll a three. You can prob­a­bly guess that the die is bi­ased. But why? One way to an­swer this ques­tion is that, in most of the situ­a­tions where one is handed a die and rolls it five times and gets the same num­ber five times, the die is bi­ased. You are ig­no­rant of ex­actly how of­ten this is the case, though. It could be the case ev­ery 9 out of 10 times this hap­pens, or per­haps ev­ery 8 out of 10 times. Now sup­pose you knew that it was 9 out of 10 times. Then it would still be ran­dom whether you are in one of the 9 cases, or in the tenth.

Next up: Refer­ence Classes for Randomness