# Coin Puzzler



## GreenMamba (Oct 14, 2012)

Let's say you're flipping a coin over and over again until one of the following sequences comes up:

1. Heads-Heads-Tails (HHT)
2. Heads-Tails-Tails (HTT)

Which, if either, 3-flip sequence is more likely to occur first?


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## clara s (Jan 6, 2014)

the world famous Penney Ante game!


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## Taggart (Feb 14, 2013)

The first.

If you get HT then another head starts a new sequence - TH is not one of the required sequences. You need another toss to get either HT or HH as your last two tosses. If you get HH then another head doesn't change anything, you've still got HH for the last two tosses; repeat until you get a T at which point you've got HHT. So HHT is more likely.


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## Stavrogin (Apr 20, 2014)

Imo, since every flip is independent, the two sequences are equally probable.

In numbers, sequence 1 has a 0.5 * 0.5 * 0.5 = 0.125 chance to happen.
Same for sequence 2.

Same for all other possible sequences, i.e.
3. HHH
4. HTH
5. THH
6. THT
7. TTH
8. TTT

overall, 8 sequences with a 0.125 chance = 8*0.125 = 1


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## GreenMamba (Oct 14, 2012)

Taggart is correct. 

Stavrogin, you keep flipping until you get one of the sequences. So those that don't end, you have to ask what the fourth flip will do.

In your above example, #3 and #5 each have a 50% chance of hitting HHT next flip. #6 is one flip from HTT. The others can't end next flip. So right there, we see that HHT is more likely to hit first.

You'd have look at further flips as well (as Taggart says, for #3, another H won't win yet, but will keep HHT viable for the 5th flip).


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## SixFootScowl (Oct 17, 2011)

GreenMamba said:


> Taggart is correct.


That is great and I found Taggart's response very clever because the info you provided was not clear on whether we were evaluating every three flips sequence separately, for example:

HTH, THT, TTH,

which leads to Stavrogin's correct answer for that condition.


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## GreenMamba (Oct 14, 2012)

Florestan said:


> That is great and I found Taggart's response very clever because the info you provided was not clear on whether we were evaluating every three flips sequence separately, for example:
> 
> HTH, THT, TTH,
> 
> which leads to Stavrogin's correct answer for that condition.


Yeah, I thought "over-and-over again" covered it, but I realize now people might think it "resets" after every third flip.

Of course, you could argue that Stavrogin's interpretation makes the question boring.


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## SixFootScowl (Oct 17, 2011)

GreenMamba said:


> Yeah, I thought "over-and-over again" covered it, ...


Good point. I did not pick up on it. Yes it is far more interesting with continuous run of flips instead of discrete 3 flip segments.

This certainly could have been a question (exactly as you worded it) on an exam for a probability course.


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