# Tossing a coin 3 times ?

1. Jul 24, 2008

### humanino

Hi,

It is often frustrating when tossing a coin if there is just one tossing and it's over. Let's say, Nicolas and George toss a coin and are after two given configurations, each his own. Nicolas likes the configuration Head-Tail-Tail. George chooses Head-Tail-Head. Whenever they argue, they play the game : they toss a coin, and the first to get his configuration wins.

Is this a fair game ?

2. Jul 24, 2008

### glueball8

why not?

3. Jul 24, 2008

### K.J.Healey

So long as it always starts from zero, I don't see why it would be any different than a single flip. Before either can possibly win, a Head-Tail pattern must occur. Then from that point, its a 50%.
Unless the wording is wrong somehow...

4. Jul 24, 2008

### humanino

I believe the wording is right. It did look pretty trivial to me too. Maybe you'd like to try it for yourself, I mean simulation on a computer, before making up you mind definitely.

This puzzle is really one of the best I have ever seen. It's simple, and nobody gets it right.

5. Jul 24, 2008

### humanino

Ah, you're right, I did not word it right... too bad

6. Jul 24, 2008

### lisab

Staff Emeritus
Hmmm, not sure if this is right...and it's hard to say with words.

George, with the HTH configuration.

It first appears that the winner is simply decided by the last toss, since they both chose HT as the first two flips in their configurations. But since the first toss of the each configuration is an H, that H could be George's winning flip, since his last flip is H.

Like I said, it's hard to say with words. English is my second language...an I don't have a first one .

7. Jul 24, 2008

### lisab

Staff Emeritus
OK, so nevermind the last post...it didn't seem quite right anyway.

So what is the correct wording?

8. Jul 24, 2008

### humanino

• I am not sure how to word it then. It would seem unnatural. They win half of the time, but when one wins he wins faster on average. I could say, every new toss comes with a new coin from your pocket, and the looser (to compensate for his loss) keeps all the coins in the end
• If I do that, you solved it already :rofl:
Congratulations lisab. You can solve even unknown problems. That is, you spotted the difference between the two configurations : one of them overlap itself, whereas the other does not. Since on a large number of tosses, they appear as often one as another, the average distance between two HTT is (8) less than between two HTH (10).

9. Jul 24, 2008

### maze

How about they just keep flipping until their pattern shows up 3 times (or something of that nature).

10. Jul 24, 2008

### chroot

Staff Emeritus
Any string of flips of length three (HHH, TTT, HTH, THT, THH, HTT, TTH, HHT) is equally probable, so the game is certainly fair. It just takes more time to play.

- Warren

11. Jul 24, 2008

### maze

Right, but if you wanted 3 repititions of hth, then you could win in 7 flips with hththth, whereas its going to take 9 flips for htt (htthtthtt).

12. Jul 24, 2008

### chroot

Staff Emeritus
The original post makes no mention of trying to obtain the desired pattern more than once; it says the first person to obtain their pattern wins.

- Warren

13. Jul 24, 2008

### humanino

The original post was wrongly worded. The problem does not make sense as such.

Contrary to intuition (at least mine), the average number of tosses until HTH is 10, whereas the average number of tosses until HTT is 8. Still, HTH wins as often as HTT. The distributions of tosses until the configuration is reached are not the same.

14. Jul 25, 2008

### dubsed

I did what you recomended and simulated this (in excel). Over 10,000 trials, both win evenly and both win in an average of 5 tosses.

15. Jul 25, 2008

### humanino

That's right
I believe that's wrong. I do not use microsoft products, so I can not help you with your macro, sorry. I have a python script, and a couple of C programs that you would need to compile.

16. Jul 25, 2008

### dubsed

Actually I did not use a macro I just used the options available in the sheet I just set up a series of tosses extracted the critical info from the seires, copied the series and info 100 times to the right (100 trials), and then copied that 100 times down (10,000) trials. At the very end I took totals and averages, averages then I can recalculate the sheet just by hitting F9. I tried to put a standard deviation to it but i dont think that is the correct way to do it. however averages are averages and pretty hard to screw up. I dont know how you wrote your program, if you were calculating odds or just running a random trials. random trials the result is prety clear and repeatable

17. Jul 25, 2008

### humanino

Just to make it clear : I have the number 8 and 10 from a researcher in statistics applied to genetics at Oxford. They are what made me interested in this in the first place. I found them by myself by writing simluation programs, but I do not believe they are questionable.

18. Jul 25, 2008

### dubsed

Here is 1/10 of the data I used. Unfortunately due to the brute force nature of the solution it was too large to keep in excel format and also in its entirety.

Description
Odd Columns - Tosses generated with =int(rnd()*2)
Even Columns - winner of series

Third to last row shows winner in odd colums and the toss he won on in the even column
The second to last row shows the toss that he won on if HTH won
The last row shows the toss he won on if HTT won

On the far right the last two rows show the following table

HTH, qty won, average toss won on
HTT, qty won, average toss won on

I know this isn't the easiest format to work with but you can import it into most spreadsheet software. If anyone can point out my error I would like to know my mistake so I don't repeat it in the future.

#### Attached Files:

• ###### 3-TOSS sample.zip
File size:
6.8 KB
Views:
137
19. Jul 25, 2008

### Howers

Huh?

20. Jul 26, 2008

### Staff: Mentor

That's only 2 repetitions of the pattern if you play it as a game. If the original intent was to allow overlapping patterns to count as multiple wins, then it isn't a fair game, but then no real game would ever work that way anyway. If it did, the best patterns to pick would be HHH and TTT, since any time you get 4 in a row, it would count as 2 "wins" and any time you get 5 in a row, it would count as 3.