A quirk of probability

Today I was playing a game on my phone. Marvel puzzle quest if you’re interested. It’s based on characters from Marvel, follows story lines and it’s game play is a bejewelled type match game. If you enjoy those games and like Marvel then it’s worth a download. And let’s face it, you’re the sort of person who spends their free time reading maths blogs. So there’s a good chance you will enjoy it!

Anyway, back to the point, at the end of each level you recieved a reward, most levels have 4 possible rewards and you get one of them chosen at random. You can replay the level and earn all the rewards, each time you get one at random, however the pool does not decrease and if you are randomly assigned one of the ones you have already won you get a non reward,  as it were.

This morning I had completed a level. I’d beaten Venom, for those of you interested. I then replayed it to try and gain the other rewards, after 6 goes I’d had the first reward then 5 non rewards. What are the chances of that? I thought, then quickly answered 1/4^6 obviously. That’s all well and good, but then I thought “it’s got to be a different one this time!” And got very angry at myself.

These are independent events. The probability of the next one being a non – reward is clearly still 1/4. Yes, the odd of getting 7 in a row are tiny (1/4^7) but so are the odds of getting 6 in a row then one of the others (3/4^7). This is something I spend a lot of time discussing in lessons, because I know it’s something learners often have trouble with, so I should have known better than to let that thought sneak into my mind.

I was reminded of the Derren Brown show where he threw ten tails in a row on a fair coin. It’s a clip I often use in lessons and ask students to conjecture how he did it, rarely do any get the right answer  (I think perhaps only once!) The answer, for those wondering, is that he threw the coin over and over until he got ten in a row,  every time he got a head he started the count again. I think he said it took him 16 hours. He could do this because he knew it would probably happen eventually.

In both these instances, this row of events taking place looks highly unlikely. However Derren repeatedly did it until he got the result he was looking for and I’m probably one of millions of people playing that game and so it’s bound to happen to someone.

Probability is a funny thing, and I think a bit of knowledge about it can lead you to fall into the trap I did. I knew the probability of 7 in a row was low, but for a second forgot that we were looking at the probability of 7 in a row given that we already had 6 in a row!