## Thursday, May 29, 2014

### Probability and St. Olaf's Lucky 13

I've posted on this before, but I've been reading off and on things about probability theory recently, and it's the sort of story that should always be brought up in considering the basics of how probability theory works. It is from Snorri Sturluson's Heimskringla. Tensions between Sweden and Norway were quite high, and a border dispute was brewing between them. So King Olaf of Norway and King Olaf of Sweden met together to figure something out:

Thereafter ambassadors were sent to Norway to King Olaf, with the errand that he should come with his retinue to a meeting at Konungahella with the Swedish kings, and that the Swedish kings would there confirm their reconciliation. When King Olaf heard this message, he was willing, now as formerly, to enter into the agreement, and proceeded to the appointed place. There the Swedish kings also came; and the relations, when they met, bound themselves mutually to peace and agreement. Olaf the Swedish king was then remarkably mild in manner, and agreeable to talk with. Thorstein Frode relates of this meeting, that there was an inhabited district in Hising which had sometimes belonged to Norway, and sometimes to Gautland. The kings came to the agreement between themselves that they would cast lots by the dice to determine who should have this property, and that he who threw the highest should have the district. The Swedish king threw two sixes, and said King Olaf need scarcely throw. He replied, while shaking the dice in his hand, "Although there be two sixes on the dice, it would be easy, sire, for God Almighty to let them turn up in my favour." Then he threw, and had sixes also. Now the Swedish king threw again, and had again two sixes. Olaf king of Norway then threw, and had six upon one dice, and the other split in two, so as to make seven eyes in all upon it; and the district was adjudged to the king of Norway. We have heard nothing else of any interest that took place at this meeting; and the kings separated the dearest of friends with each other.

This I take to be perhaps the central question in philosophy of probability: What is the probability of St. Olaf rolling a thirteen with two ordinary six-sided dice? And of course it connects to the fact that probabilities are not just magic numbers independent of everything else; it matters what kinds of events you are assuming to be possible.

1. It's obviously impossible to roll a 13 with two six-sided dice, but the scenario described above isn't normal. Six-sided dice are traditionally arranged so that each pair of opposite faces totals 7: 1-6, 2-5, 3-4. Thus, a split die which landed with both halves dots-up would always show seven. Assuming an even split, the odds of showing thirteen are 1/6 * 1/2 * 1/2, or 1 in 24. A natural 12 is actually much less likely, at 1 in 36.

The only actual question, then, is how likely is it for a die to split?

I think here the event we assume possible is a die (or even two) splitting and from there we study the probability of the combination including the split.

A somehow unrelated question, if I may : is St Olaf here the same one from the story of St Dunstan?

3. branemrys3:54 PM

He lives in about the same time period, but I didn't explicitly mention him in talking about St. Dunstan. He was a relatively minor king who was defeated by Cnut, whom I did mention, though.

4. Delightful story for one, like me, who has a particular attachment to 13!

5. branemrys12:30 PM

It's one of my favorites!

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