Nobody Wants to Believe in Chance

Nobody wants to believe in chance. Luck, maybe. Chance, no. Everybody wants there to be meaning, purpose, process, cause, reason, design, whatever... explaining why things turn out as they do. Rejection of chance is everywhere. But nowhere more so than in the casino, the sanctum sanctorum of chance incarnate.

I'll use a video poker example to show how strongly solid citizens seek methods behind the madness. I could cite other games, but video poker comes to mind because of an argument I stumbled upon last week about how draw poker machines pick cards to replace those thrown away after the initial deal.

Andy asserted he "read somewhere" that machines picked 10 cards when you hit the button, displayed one through five, and put six through 10 -- in order -- underneath. "When you toss a card," he maintained, "you get what's already in back of it."

Sandy's theory, based on something a slot attendant told her, also was that machines picked 10 cards at once and exposed the first five. But replacements came from the stack one at a time, as needed. "If you get rid of three," she insisted, "your new cards are the sixth, seventh, and eighth of the original set.

Randy was sure both of these were wrong. He'd heard someone on the bus saying machines pick five for starters, then get as many as needed for the draw -- one at a time -- when you ask for them.

They wanted me to tell who was right. Instead, I asked, "what difference does it make?" They all replied to the effect that the method determined the outcome. All else being equal, hands end up different each way. "Ah!" I realized. "You hoping for something that makes you win or lose. You're denying chance." I told them that, in honest games with cards picked at random from virtual decks that were perfectly mixed and arbitrarily cut before each round, the approaches are equivalent. And, if the games were rigged, any of these -- or lots of other -- means could be used.

"The key," I went on, "is that these methods -- and alternates you could devise -- all offer the same chance." I posited getting 2-H, 3-H, 6-C, 8-H, 10-H in a jacks-or-better game. Anybody would hold the four hearts and try for the flush. "You can find the chance of making the hand without knowing how the replacement for the six of clubs was picked," I contended.

"A full deck has 52 cards, and five have been removed so 47 are left," I continued. "There are 13 hearts, but you already have four so nine are left. This means the chance that the replacement for the six of clubs will be a heart is nine out of 47." A little long division on a napkin showed 9/47 to be just over 19 percent.

"Yes, but..." the trio chimed in unison, "you would have gotten a different card with each of our methods." This was true, I admitted. "But the actual card you draw is where luck rears its head. What drives the game isn't the hand you get, it's the chance of various consequences when you act."

I proposed they run an experiment at home to convince themselves. You can do it too. Draw three columns -- A, B, and C -- on a sheet of paper. Shuffle a deck and deal five cards face-up. Put five more cards above them, face-down. Now play, trying all three ways of drawing to replace discards.

First, flip the corresponding face-down card or cards and write the value of the final hand in column A. Second, dump the first draw and substitute as many cards as needed from the top of the deck, marking the value of the result in column B. Third, ditch this draw and take replacements from anywhere you want in the deck, noting the value of the outcome in column C. Put a star in the column with the highest poker value.

Shuffle and repeat the game. You'll get different ultimate hands on each round for A, B, and C. But it won't take long to see that the stars are scattered around, one method being as good as the next. Which confirms the caveat of the poet, Sumner A Ingmark: