# Rank Imbalances (alternative version)

“Well technically we got our empty column back,” said Haw.

<sarcasm> A fat lot of good that did </sarcasm>,” replied Hem, “seeing we had to lose it immediately.”

I enter the card room and survey the current game state.

“Allow me to introduce ourselves,” says Haw. “We’re the Little People – Hem and Haw from the short story Who Moved My Empty Column?

“I’m Spider GM,” I reply. “I think I know you already – after all I’m the writer of this blog.”

My other students introduce themselves to the Little People. I’ve watched Hem and Haw play before, and it seems they are decent enough players but prone to going on tilt when things don’t go their way.

“We’ve just dealt a fresh row of cards,” I say. “Before making a move I want you to evaluate the position. Do you think we are going well, badly or somewhere in between?”

“Could be worse,” says Hem. “At least we can get back our empty column.”

“But what do we play after getting back the empty column?” asks the Lion.

“Well we can also expose a card in column Three” says the Eagle.

“Uh oh,” says Sand Griper. “I think the laws of probability are rigged.”

The Sandgroper is not one of my better students. He got that nickname because he always likes to spend a lot of time complaining about his bad luck – time that could be much better spent on learning statistics 101.

“Why are the laws of probability rigged?” I ask.

“There are 49 cards exposed. I see six Jacks but only one Ten. This is remarkable – surely that shouldn’t happen very often assuming perfect shuffling.”

The Sand Griper clicks his tongue and Ninja Monkey immediately grabs two decks of cards and deals 49 cards face upwards. He rinses and repeats for 10,000 trials. It takes a mere six seconds to tally the number of remarkable deals according to the Sand Griper’s definition.

“I think the Sand Griper may have a point,” says Ninja Monkey. In only 61 trials did I get 49 cards with at most one Ten and at least six Jacks.”

“Not so fast,” I reply. “How many games did you play?”

“Also why did you choose Jacks and Tens? You obviously chose them because of the current game state. But you might have chosen Threes vs Fours or Queens vs Kings. For your reasoning to be valid you have to nominate Jacks vs Tens before dealing a hand.”

The Sand Griper starts squirming – and with good reason.

“Alternatively, you can say that a set of 49 cards are rigged if there is ANY pair of consecutive ranks X and Y (such as Threes vs Fours) such that we have AT MOST ONE of X and AT LEAST SIX of Y. Also, remember that X can be Y – 1 or Y + 1.”

I click my fingers. After six seconds of dealing and shuffling Ninja Monkey tells me out of 10,000 trials there are 1251 satisfying the above conditions.

“Therefore, if you play 10000 games you should get a remarkable deal 1251 times.” Since you played 10 games you should get a remarkable deal 1.251 times. Now you told me you played about ten games and you only complained about getting a remarkable deal once. So perhaps there is nothing remarkable about this after all.”

The Sand Griper continues to squirm.

“Now, going back to the lesson …” I continue.