The Curious World of "Chances" in Games

If we were to sort games according to “how much does chance play a role in deciding the winner” we can place all games on a spectrum, with “Chess” being leftmost – not playing any role – and “Throwing Dice” where it’s up to the universe which side comes up.


In Chess, players take turns at moving one figure at a time (two while castling) on a 8x8 board consisting of 32 black and 32 white squares. Players decide which piece to move next and all the information needed to make this decision is laid out in front of them – there is no luck involved.

Playing Chess means running through a lot of possible states of the board, a couple of moves in advance, and playing the first move of that variation. Now it’s your opponent turn and the only way you can be “lucky” in Chess is by having your opponent “not see” a move, after which you gain some advantage.

Now let’s say you’re a perfect Chess player. You can calculate 50 moves in advance in a matter of seconds and play the best move. But when paired with a player of the same class – you will never win. The outcome of a perfectly played Chess game is draw.

Take a second and think about that. For all we humans know at the present moment, and for all computers can calculate, there is no way to “force” a win in Chess – only hope your opponent will make a mistake.

And hoping that your opponent will make a mistake is what most people call “luck” which is a concept of chance.

Texas Hold ‘Em

Hold ‘Em is the most popular version of the card game Poker. Players are dealt two cards each and the game begins. Throughout the game, players take turns in placing bets which generally correspond to what they think it’s the “strength” of their hand – a combination of the two cards sitting in their palm and the open cards on the table. When the hand-cards are opened, at the end of the game, the player who has the most improbable combination of cards wins.

Playing a round of Hold ‘Em means calculating the probability that the pair of cards you’re holding away from other players’ view, when combined with the open cards on the table, will make a combination that’s “stronger” than opponents’ combinations (while trying not to get too dizzy from the whisky and cigarettes).

If you watch Hold ‘Em on TV, you’ll notice each hand has a particular probability of winning assigned to it. Often, when two players are left on the table, one of them has “100%” written next to his hand. He’s surely a winner, but why do they continue playing? Why isn’t the game over? It’s obvious, of course, that the winning player lacks the information we have about the other guys’ hand, so according to his perspective – there’s still chance that he’ll lose.

So “chance” in card games is nothing more than we not knowing the other guys’ hand or we not knowing the card at the top of the pile. After the deck is shuffled, most card games become truly deterministic.

What does “chance” mean when cards are played open?

Throwing dice

The last example I want to discuss is “Throwing Dice.” Throwing dice is what most of us consider “really really random”. You take a number of dice, you shuffle them in your hand and throw them on a flat surface.

I made a variation of the game, it’s called “Placing Dice”. You take two dice, pick which side you want to point upward and you carefully place them on a flat surface.

Or I have another one: you take two dice, rotate them as you wish and drop them from a 0.5cm distance on a flat surface. Most of the time, they land as you wanted them to.

“Throwing Dice” is really just a more aggressive version “Placing Dice.” And what we call a “chance” in “Throwing Dice” is really just inability to precisely control our muscles, or the wind, or have in mind the slight bumpiness this flat surface has. Just as in Hold ‘Em, chance is just not knowing enough.

That’s what I find really interesting.