What Are The Odds?

Being of a mostly philosophical bent, the HP doesn’t do numbers well; there are just too many of them. Oh, sure, back in high school he liked algebra and trigonometry because they seemed like puzzles, but he hit the wall when it came to higher math – calculus was, and still is, a mystery. 

A general discomfort with numbers is why he didn’t go into accounting, engineering, nuclear physics, etc. He's much more comfortable with words. 

But that said, and quite ironically, the HP often has numbers on his mind. 

Bridge involves numbers

I play a lot of duplicate bridge. At each table, there are four players, four suits, 52 cards in the deck, thirteen cards in each hand. We typically play 28 deals in a session, and it lasts about 3.5 hours. On each deal we count how many tricks we can win, figure the odds for taking a finesse (50% more or less depending on what we intuit from the bidding), calculate the chances of “setting” the opponents if we let them play the hand, and compute differential scoring for being on offense or defense. We try to remember every card played on each trick and tally the number of trumps and other key cards remaining.

The HP can usually handle these calculations, but there are other, scarier numbers at work in bridge too. For example, the odds of holding a “Yarborough” (a hand with no card higher than a 9) are 1828 to 1. The odds of holding a “perfect hand” (one that can take all 13 tricks in Notrump regardless of what the other players hold) are almost 170 million to 1.  

And get this: there are 53,644,737,765,488,792,839,237,440,000 possible different deals. That’s more than 53 octillion (5.36 x 10²⁸).

For all you chemistry majors reading this, that’s more than “Avogadro’s number” (the number of elementary particles per mole of a substance). Poor Avogadro – his famous number is a mere 602 septillion (6.02 x 10²³).

The enormity of the number of possible deals can be appreciated by considering that if each deal occupied just one square millimeter, you would need an area more than a hundred million times the entire surface area of the Earth to hold them all.

Numbers and rare events

Unrelated to bridge, and on a more practical level, the HP has been thinking about the Law of Large Numbers and the probabilities of certain rare events. For example, if the chance of something happening is 1 in 1,000 and you repeat the event 1,000 times, what is the likelihood that it will occur at least once during those 1000 occasions? It turns out the answer is 63%.

Don’t ask me to explain why—I don’t do numbers, remember?—but it makes sense intuitively. And if you understand calculus, apparently this equation proves it:

That percentage is good to know. Consider the odds of falling down a flight of stairs. They’re estimated to be roughly 1 in 20,000. Not a big deal, right? Well, maybe not on any given day, but consider how many times you go up and down stairs in a lifetime. More than 20,000, right? According to the above equation, there’s a 63% chance you’ll fall on one of those trips. (Pun intended.) According to the CDC, falls cause nearly one-third of all non-fatal injuries, and more than 800,000 of those injuries require hospitalization. Even more surprising, about every 20 minutes an older adult dies as the result of a fall.

These statistics were a reason I persuaded my homeowner’s association to let the HP install handrails on our exterior stairs. And considering the number of times our residents traverse those steps in a year, the Law of Large Numbers was bound to catch up to us at some point. It still might, of course, but at least now there’s a handrail to grab onto. 

Without the ADA-compliant handrail attached to the 2” x 8” banister on the left, there was no safe way to catch oneself in the event of a fall.

It’s a small world

In a lighter vein, odds and probabilities seem to come into play often in the form of “small world” stories. For example, when talking to the guy across the hall in my condo I learned that he is also from St. Louis, had lived in the subdivision next to ours back there, went to the same high school my kids did, and was just one year behind my younger son, whom he knew casually. So of course I wondered, “what are the odds of that?”

Then a new couple moved in, and while talking to them I found out they had just moved from the apartment building I lived in before I bought the condo. Further inquiry revealed that they lived in the exact same unit I did, one of more than 50 in that other building. What are the odds of that?

Then I learned that the guy downstairs from me is a Navy diver. He knows another Navy diver who is the son of one of my law school classmates. What are the odds of that?

And one day I met a new player at the bridge club, got to talking to him about our common interest in health law, and gave him my business card. A few days later I saw him again and he began with, “I know someone you know, and in fact I’m married to her.” It turns out he took my card home, laid it on the table, and when his wife saw it she said, “I went to college with Stuart! How the hell do you know him?” Susan and I hadn’t seen each other in more than 50 years, but we’re now friends again. Small world, huh! What are the odds of that?

Everyone has small world stories, and they’re intriguing because the odds of them happening seem astronomical. But since we all have these stories, the odds must not very high after all. And they aren't, actually.

We all know or have known hundreds, maybe thousands of people in our lives. And each of them knows hundreds or thousands of others. The “six degrees of separation” phenomenon comes into play. (That’s the premise that every person is connected to every other person on the planet through at most a chain of six common friends or acquaintances.)

So when two strangers meet—at a bridge club or on an airplane, for example—it’s virtually certain that they have a common acquaintance. What’s really against the odds is for them to discover the connection. And it’s intriguing to imagine the scores of “small world” stories that go unrecognized in the interactions we have with people every day. (More on the small world phenomenon can be found in a scholarly article at http://www.appstate.edu/~hagemansj/smallworld.html.) 

And finally

If ever the HP is dissatisfied and ponders what it would be like to go back to his youth, all he has to do is think of calculus. At this age, those are the kinds of numbers that just aren’t worth trying to understand.

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