Note: Not at the old Poker1 site. A version of this entry was first published in Card Player. This entry in the "Aunt Sophie" series covers pan (or panguingue), which is a multi-player form of rummy, often played for money.
Aunt Sophie feels the weight
As I left the Anaheim Club, a fascinating sight accosted my eyes. In addition to the rising sun to greet me, there was my Aunt Sophie in a lavender jogging suit engaged in — what else? — jogging. Now some people look good in jogging suits, even if they are the velour type that would never stand up to actual use for their purported purpose. To say that Aunt Sophie is one of those would be stretching things a bit.
“Aunt Sophie,” I greeted her, “you had no way of knowing the game would just have broken up, so running into you here must truly be a coincidence. But tell me, whatever are you doing?”
“Dollink,” she began, “that Dessert of the Month my Cousin Moishe gave me as a gift for my birthday is killing me. Last month it was Chocolate Decadence. This month it’s candied pears in Cointreau. Who knows what it will be next month? Apple Pan Dowdy, or maybe Armenian Halvah, or Crepes Souffles au Citron, or Diplomate au Kirsch, or possibly Caribbean duckunoo. Whatever it is, it will put more weight on me. And that I don’t need. So what I’m doing is exercising to lose the weight those desserts have been putting on me. And I hope you’ll come over once a month at least and eat the whole thing. Right now I’m on my way to the Fitness Center, which just happens to be across the street from this fine establishment, for my aerobics class. It’s actually not starting for another hour, so let’s go back in and I’ll buy you breakfast.”
I wasn’t sleepy yet, so I readily agreed. We went to the coffee shop. Steak and eggs with home fries for me, and steaming coffee. Just dry toast for Aunt Sophie.
“Why,” she demanded, “does so much misinformation get spread in print about pan?”
“Well,” I temporized, “perhaps because so little correct mathematical analysis has been made of the game. Much has been written about the scientific playing of poker because a lot of people need the information, since professional poker playing is becoming more and more socially acceptable. Pan is still viewed generally as a recreation, and, since it’s so hard to beat on a regular basis, there are not very many professional pan players. For that reason, accurate information about the game is not so much in demand as for other games. But, before I continue on my soapbox, what brings on the question?”
“Recently,” she continued, “a college professor wrote a letter to the editor of The Card Player that supposedly gave the true odds on putting out a pan hand. He made this interesting statement: `Each player has an equal chance of going out regardless of the number of hits needed (luck is bigger than probability in pan).’ If that’s true, how come some players put out so many more hands than other players? If that’s true, how come I’m supposed to play the way you’ve been teaching me?”
“That,” I opined, “is an interesting question. I noticed that letter myself, partly because of my interest in correct usage of the English language. He stated that `the odds for each player are one divided by the number of players,’ and went on to propose that the odds for one player going out are 1, for two players are 1/2, for three 1/3. He skipped the ones in between, and said for seven they are 1/7.
“Now, my grammatical objection is the use of the word odds. Odds refer to the likelihood or unlikelihood of a particular event, and are always expressed as some number to some number. For example, if you thought that the Red Sox were twice as likely as the Mets to win the World Series, you would have said that they were 2-to-1 favorites. If you wanted to describe the unlikelihood of the Mets winning, you would have said it was 2-to-1 against their winning. Odds are nearly always expressed as one whole number versus another, sometimes a mixed fraction against another. That is, you could say 3-to-2 against something happening, or one-and-a-half-to-1. Those numbers are sometimes written separated by a colon, but they are still read as if the colon was the word to. So 2-to-1 could also be written 2:1, and 3-to-2 could be 3:2. But if someone reads those figures aloud, he still says two to one and three to two.”
I paused to eat some of the sirloin. “Chances on the other hand always refer to the likelihood of something happening, and are always expressed as a fraction. One fraction, less than 1, except in the case of something being a 100% certainty, in which case the chances are precisely 1. So, if something has a 1 in 4 chance of occurring, this can be expressed as 1/4, 0.25, or 25%. Notice that if two figures are used, the word in always goes with them, or, sometimes, out of. You could also say the chances are 1 out of 4. The word to always goes with odds. If two numbers are used, in the case of chances, the second figure is always the total number of events.
“In the case of odds, the higher figure always comes first, and it is the difference between the total and the second figure. So if there is a 1 in 4 chance of something occurring, there are also 3-to-1 odds against it occurring. See the relationship between the figures? First you need to know the odds. You express that as a fraction. Then subtract the numerator from the denominator, invert the result, and you’ve got the odds. For example, we’ve been talking about chances of 0.25. As a fraction, that’s 1/4 (1 over 4). Subtract 1 from 4, and you get 3. Invert it, and that’s 3/1, more commonly expressed as 3-to-1 or 3:1. That’s a very common mistake, and you frequently see it in promotions for supermarket, breakfast food, and airline sweepstakes, and by lottery officials.”
“My goodness,” exclaimed Aunt Sophie, “where do you learn all this stuff?”
“I got it from an elementary statistics courses at Stanford,” I responded, “but any true mathematician knows the difference between odds and chances. Just to complicate things slightly, of course, chance and chances are often used interchangeably. Odds, however, is always plural.”
I caught the waitress’s eye and pointed at my coffee cup. “Misuse of technical terms by a supposed mathematician is not the big problem, of course. Incorrect pan information is the subject here, and I’ll just give a small example that shows how wrong it is. According to the letter, if one player plays a pan hand — and when would that happen, except maybe in pan solitaire while waiting for a game? — ‘the probability of putting out the hand is one because, being the only player, he can take as many picks as he needs.’ Yes, and what if this player has a true Yarborough? A hand that has absolutely no matches, and no cutoffs. He can pluck at that deck the rest of his life, and he’ll never put the hand out. That’s an extreme case, of course, but it means that even with only one player in a hand it’s still not a 100% possibility of the hand being put out. Probably the chance is somewhere around 0.99. So let’s extend that to two players. One has a Yarborough, and the other any better hand. Obviously the chances are not the same for both. The player with the Yarborough has no chance of going out, while the other player now has precisely a 100% chance. This assumes of course that the mucker would just keep shuffling up the cards as they were used up. And, again, with three players. Let’s again say that one of them has that hand that absolutely cannot be put out, and the other two have hands that are approximately equivalent. Now it seems that of the three players, one has no chance of going out, while each of the other two has about a 1 in 2 chance (not 1 in 3 as the letter writer would have us believe.”
“Sure,” Aunt Sophie object, “but how often does a player get a Yarborough?”
“Rarely, to be sure,” I responded. “I was deliberately picking an extreme case to show how there are no absolutes in pan. But it certainly is true that some hands are better than others. You could be dealt 10 cards flat, three pairs, maybe, with nearly a third of the remaining deck able to put you out. Maybe I played a real pisser. I need five hits, and they’re gut shots to spade runs, or comoquing valles. Maybe 10 cards remain in the whole deck to put me out, or something in the order of 1/30th of the deck. Obviously if we played this contest out hundreds of times, you would put your hand out far more often than mine. So you can see that even though there are two of us, the chances are not the same for each of us. Forget about luck. That evens out in the long run. In the same long run, the players who play the best hands are going to put out more hands, and win more. And so my advice about what hands to play, and how to play them, and in what situations to play them, still holds. And remember, you can’t believe everything you read.”