Essay of little or no significance #1: probability

I have just rolled a dice. What is that probability that I have rolled a 5? Is it one in six? No.

Before the roll the probability of rolling a 5 was one in six. That’s because a 5 is just one of six possible outcomes all of which are equally likely. But what about after the roll? If I have rolled a 3, is it equally likely that I have rolled a 5? Of course not: I have rolled a 3, I have rolled no other number.

The probability that I have rolled a 5 must therefore be one in one or zero in one, not one in six: either I definitely have or I definitely haven’t rolled a 5.

I'm not usually one to indulge in "possible worlds", but here they can be used to great effect. Imagine that I am about to roll a dice, but stop me; freeze the frame. Duplicate the scene in your mind five times and play all six possible worlds at once. It's feasible that in each possible world a different number is rolled. There was, after all, a one in six chance for each number.

Now rewind the tape. This time, allow me to roll and cover the dice before you freeze the frame and duplicate the scene. Play all the scenes at once. You will find that in every possible world I roll the same number. If I rolled a 5 in the real world, I will have rolled a 5 in every other possible world. The probability that I had rolled a 5 was not one in six but one in one; there was no chance that it could have been any other number. As they say, alea iacta est!