Dragons in the Algorithm
Adventures in Programming
by Michael Chermside

How Wierd Is That!

On reddit, I recently answered a question which I thought was rather interesting.

There are 8 cards to be drawn from. We are 5 players, each person picks a card randomly from this pile.

The first time around, everyone picked cards at random. After the game was over, we shuffled all 8 cards together and set the game up again.

In both rounds of the game, me and the other guy picked the same cards.What is the probability of this happening? Or am I reading too much into this and the two events are unrelated and it was just a coincidence?

A naive answer would be that in the second hand there is a 1/8 chance of "me" getting the same card as last time (since there are 8 cards), and with the 7 cards left "the other guy" has a 1/7 chance of picking the same as last time, so the probability is 1/56 or about 1.8%.

But that's deeply misleading.

This is actually a very common intellectual fallacy when dealing with probability. After all, wouldn't the poster have been just as surprised if he and a different player had gotten the same cards twice? Or even if any 2 players had gotten the same cards twice? The probability of some PARTICULAR weird thing happening is usually quite low, but the probability of SOME weird thing happening is much higher. People calculate the former and comment on what an amazing coincidence it was. It's like someone drawing a 6 of diamonds and saying "Amazing! There was only a 2% chance of getting that particular card!" It's true, but highly misleading.

The physicist Richard Feynman is reputed to have made a joke about this. He walked into a lecture and said:

You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!

At any rate, a much better answer for person would be the probability that some pair of players in the game would have gotten the same cards twice. The chance of that happening is 107/840, or more like 13% -- much more likely. And do you think they might have been surprised if they had gotten the opposite cards as the last time? Or if everyone had gotten cards in numerical order? Or reverse order? The chance that something would have surprised this person seems very large.

Posted Tue 03 May 2011 by mcherm in Math