There’s this internet show called Scam School. It’s a show that shows little puzzles and games you can play in bars or at parties to win drinks and amaze you’re friends/enemies. Anyway, the most recent episode features a card based guessing game w/ some (I guess) counterintuitive statistics involved. If you watch the video you won’t have to read the next paragraph where I explain the game.

It works like this. I bet you, giving you very good odds, that you can’t guess the next card in the deck *wrong* all the way through. So, you have to guess the card value, not suite, and be wrong 52 times consecutively. So, if you say 3 and a 3 comes up you lose. You can change your guess every card. The game usually is done so you can only see the most recently displayed card, not the whole history of cards shown. You would probably lose the bet because the odds of making it through the whole deck is much lower than people would (I guess) assume.

In the show he talks about how the stats are assuming no strategy and then he kinda asks people to figure out what could be done with a strategy. I was awake and intrigued so I made a spreadsheet with the stats with the best possible strategy. I think I got it right. Ya’ll can check my work.

The strategy is simple. Guess whatever card has shown up the most. So, for the first guess it’s random. Then guess whatever the first card shown is until some card is shown twice. Then guess that until some card is shown thrice… and so on. Once a card has been shown 4 times you win by just guessing that card. So, in the spreadsheet I calculate the odds of their being a duplicate, triplicate, or quadruplicate at each point in the deck, and I calculate the odds of guessing wrong at each point of the deck giving each possible duplication situation, and then I merge all the probabilities. That gives the odds of guessing wrong at each point in the deck. To get the cumulative odds of getting that far into the deck you just have to multiply all the odds of getting to each prior point.

I posted this to the forum and was slightly corroborated by someone, however someone else was saying they made a brute force program that was getting something like a 27% success rate, where my statistics, as you can see, predict only a 19.5% success rate. So, I whipped up my brute force simulation program. There’s a .rar containing the .exe available here. It’s not very pretty or anything, but it corroborates my statistics, so I’m feeling pretty confident at this point.

So that’s what I did recently. Take that people who think I’m wasting my life.

Actually, I have thought of one inaccuracy. Technically the stats in the spreadsheet are a bit off because it calculates the odds of you getting a duplicate, but doesn’t factor in the odds that that duplicate would cause you to lose, like in the case that the first two cards are the same. I’m not sure how big a difference that would make, but seems like the simulation should avoid that problem. Currently it looks like the error results in an approximately 3% overestimation of success.

If you actually look at any of this stuff and find an error (or know how to fix that one for the spreadsheet), or if you want my source code or anything for the simulator, please let me know.

Update: Well, my updated simulator (link auto updates to the newest version) and now it the success rate seems to hover around 27% which doesn’t seem to match the statistics. I dunno what gives.