More Probability – HTH vs. HTT
Ξ May 8th, 2009 | → 0 Comments | ∇ Maths, Oxford |
Imagine tossing a fair coin successively, and waiting until the first time a particular pattern appears; say HTT. For example, if the sequence of tosses was HHTHHTHHTTHHTTTHTH, the pattern HTT would first appear after the 10th toss.
Ok, now let’s take two such patterns – HTT and HTH. Given both these sequences, and a lot of trials [where you conduct this, "it first appears when" experiment, and then average the number of tosses], is it more likely that you’ll:
- hit HTT in less tosses than HTH;
- hit HTH in less tosses than HTT;
- find that the number of tosses is the same?
Most people [many mathematicians amongst them] will pick the third option. Surely, any such pattern is equally likely to show up in some yet to be discovered average number of tosses!
Actually, it’s not the case – that they’re equally likely. In reality the average number of tosses required to see HTH is 10, whilst for HTT it’s 8! How can that be???
Let’s see why.
Note that HTH overlaps itself, i.e., if you got HTHTH you’ll find that you’ve got two occurrences of the pattern in only five tosses, i.e., HTHTH and HTHTH. Ah, so doesn’t this sound like HTH is more likely then, rather than the other way around?
Well, with HTT there isn’t such an overlap – and it turns out – perhaps unintuitively – that that’s important; in a way that leads to HTH’s downfall. So, let’s run a couple of experiments to see how this works.
Let’s go looking for HTH
Best scenario:
Toss Result Comment H 1st token in our pattern excellent start! T 2nd token quite excited! H 3rd token We won!
Second best scenario:
Toss Result Comment H 1st token in our pattern excellent start! T 2nd token quite excited! T Bugger!
Now we’ll need to continue tossing the coin until we see an H; as that’s the first token in our sought-after sequenc
Now let’s go looking for HTT
Best scenario:
Toss Result Comment H 1st token in our pattern excellent start! T 2nd token quite excited! T 3rd token We won!
Second best scenario:
Toss Result Comment H 1st token in our pattern excellent start! T 2nd token quite excited! H/H Bugger!
However, and this is the important bit, at this stage we don’t need to toss the coin again in order to get to find our starting token – we just threw it – an H!
If you doubt any of this, here’s a little simulator I wrote [CoinToss.zip contains CoinToss.exe].