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Monday, March 4, 2013

Good news! People are smart! Kind of!

For customer HAROL we wrote a little memory game. There are 16 cards (8 pairs), and you get 3 attempts to try and find a match. If you do find a match you might just win an iPad! If you don't - well, you still get a discount coupon. The game was played on iPads set up at Batibouw. There is no online link for the game, or I'd link you there *.

Before we sent the game off to Batibouw, I sat down to calculate the likelihood of finding a match within your three attempts: it shouldn't be too hard or Harol would never get rid of its iPads. For calculations, I would assume a 100% perfect memory. The most interesting part here is how close people would get to that 100% perfect memory score. Are people smart? Kind of? Or are they really dumb?

I haven' t had a lot of formal statistics training, so I've largely relied on The Big Three: "simple common sense", "blatant disregard for other people's opinions" and "shameless guessing". Readers with a background in statistics: feel free to set me on the right path in the comments. I'll feel free to blatantly disregard you.
Anyway, here's what I figured:

Three turns, six cards. Rather than calculate the 'winning' percentage, I'll calculate the 'losing' percentage, and deduct from 100 %. Losing is the one route you take through the game where you don't find a match, winning is any of 3 distinct events - maybe even 5.
  • Card 1: nothing to do here
  • Card 2: Lucky guess chance to win. Chances to lose: 14/15
  • Card 3: May be a match for one of the 2 previous ones. 12/14 it ain't though.
  • Card 4: And if it ain't, there's still a lucky guess chance: 12/13
  • Card 5: A match for any of the previous ones is starting to be likely, but not quite: 8/12
  • Card 6: Last chance to hit a match: 10/11
So where does that put us? Far as I can see, we just multiply all of those fractions, and come up with the losing percentage: 0,4475524475524476. Say 45%. You're actually 55% likely to win if you play it smart. (If you just guess blindly, your winning chances are obviously (1-pow(14/15, 3)) = 18.6 % = not looking good).

So how did people do?

**** drumroll ****

Only 6% away from a perfect memory.

Good news! People are smart! Kind of!

* For those of you who care: it was an MVC app, some nice ajax, css transitions that I could afford to rely on, as it only needed to work on iPad. Who knows maybe it will be recycled for another promotion that will run online.


  1. Sooo. Turns out I was wrong. I was pointed to a mistake I made in the card 4 step (thanks Machteld!):
    There is an option other than a lucky guess: uncovering a match to card 1 or card 2. That would solve the game for the third turn (card 5 and 6). This possibility is not relevant for card 2 (no previous cards known) and card 6 (no turn left to play your known match).
    The (hopefully!) correct winning probability is
    1 - ((14/15) * (12/14) * (10/13) * (8/12) * (10/11)) = 1 - 0,372960372960373 = 63 %

    I guess the world population just got a little dumber.

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