Odds and the transitive property

I got this from Mano Singham’s blog: he is a physics professor who mostly writes about social issues. But on occasion he writes about physics and mathematics, as he does here. In this post, he talks about the transitive property.

Most students are familiar with this property; roughly speaking it says that if one has a partially ordered set and and then . Those who have studied the real numbers might be tempted to greet this concept with a shrug. However in more complicated cases, the transitive property simply doesn’t hold, even when it makes sense to order things. Here is an example: consider the following sets of dice:

What we have going here: Red beats green 4 out of 6 times. Green beats blue 4 out of 6 times. Blue beats red 4 out of 6 times. All the colored dice tie the “normal” die. Yet, the means of the numbers are all the same.

Note: that this can happen is probably not a surprise to sports fans; for example, in boxing: Ken Norton beat Muhammed Ali (the first time), George Foreman destroyed Ken Norton and, Ali beat Foreman in a classic. Of course things like this happen in sports like basketball but when team doesn’t always play its best or its worst.

But this dice example works so beautifully because this “impossibility of the dice obeying a transitive ordering relation is theoretically impossible, by design.

Movies
Since the wife has been gone on a trip, I’ve watched some old movies at night. One of them was the Cincinnati Kid, which features this classic scene:

Basically, the Kid has a full house, but ends up losing to a straight flush. Yes, the odds of the ten cards (in stud poker) ending up in “one hand a full house, the other a straight flush” are extremely remote. I haven’t done the calculations but this assertion seems plausible:

Holden states that the chances of both such hands appearing in one deal are “a laughable” 332,220,508,619 to 1 (more than 332 billion to 1 against) and goes on: “If these two played 50 hands of stud an hour, eight hours a day, five days a week, the situation would arise about once every 443 years.”

But there is one remark from this Wikipedia article that seems interesting:

The unlikely nature of the final hand is discussed by Anthony Holden in his book Big Deal: A Year as a Professional Poker Player, “the odds against any full house losing to any straight flush, in a two-handed game, are 45,102,781 to 1,”

I haven’t done the calculation but that seems plausible. But, here is the real point to the final scene: the Kid knows that he has a full house but The Man is showing 8, 9, 10, Q of diamonds. He knows that the only “down” card that can beat him is the J of diamonds but he knows that he has 3 10’s, 2 A’s. So there are, to his knowledge, cards out, and only 1 that can beat him. So the Kid’s probability of winning is which are pretty strong odds, but they are not of the “million to one” variety.