LOOKING ON THE BRIGHT SIDE
By Denny Teichow,
Do you like to play cards? Lots of people do, but I don’t think cards are the ubiquitous entertainment that they used to be. My parents and most of their friends gathered at someone’s house to play cards at least once a week all year round. But that was B.TV. Most card players think they have a pretty good understanding of how frequently certain card hands will show up during a deal. But as writer Artemus Ward observed, “It ain’t so much the things we don’t know that gets us into trouble; it’s the things we know that ain’t so.”
Suppose someone shuffles a deck and deals you five cards. Which of these hands are you LEAST likely to receive?
If you chose the upper left set, then you are normal. You are also wrong.
All these sets are equally likely to be dealt.
I know; there’s only one royal flush in spades in the whole deck, and there are a LOT of sets like the one on the upper right. But “like” is not the same as identical. How many sets of five cards are there in a deck that are identical to the one on the upper right? Yup, just one. There is only one set of cards in the deck exactly like the one on the upper left, and there is only one set of cards exactly like the one in the upper right. These two sets are equally likely to occur. And, of course, the same is true for the third set. You might want to pause here to let that sink in.
When thinking of cards we have a tendency to think about groups more than individuals. The group of royal flushes in a deck of cards is a tiny group, just four members. Whereas, the group of “junk” hands, like the one on the upper right, is a huge group, over 2 million of them. If you have been paying attention in the school of life, you should know by now that a statement that may be true when comparing groups, is not necessarily true when comparing individuals from those groups. That’s true of cards, as well.
If the question above had been “Which is least likely to be dealt, a hand from the group of royal flushes, or one from the group of ‘junk’ hands?”, then the correct answer would be, of course, the royal flush group. But the original question was about individual hands, not groups.
A friend of mine stumbles over this mental mistake when buying a lottery ticket. He thinks that when picking lottery numbers, you should never pick consecutive numbers, like 16, 17, 18, 19, 20. His reasoning is that there are far fewer sets of numbers like that than there are junk sets (no pattern), like 16, 23, 26, 32, 41. This is true, of course, but he makes the mistake of comparing groups when he is betting on an individual. For the numbers 1 – 45 there are only 41 sets of five consecutive numbers, and there are over a million junk sets. But, again, “like” is not the same as identical. How many sets of five numbers are exactly like this: 16, 23, 26, 32, 41? Yup, only one set exactly like that. A set of consecutive numbers is just as likely to come up as any individual junk set. When you buy a lottery ticket you are not betting on the group of all junk sets; you are betting on one specific set. My friend says that a set of five consecutive numbers has never come up in the lottery. True. It is also true that there are over a million junk sets (almost all of them) that have never been a winner. If lotteries go on long enough, eventually the ping pong balls will spit out a consecutive set of numbers, and that will happen before many of the junk sets have been winners.
I am tempted to remind you of how this same lesson applies to people (individuals versus groups) and how we make the same kind of mistakes in thinking about people, that you did on the questions about cards or lotteries.