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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 comment:

  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.