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No preamble!
New coin game, you vs me.
Step (1)
Each of us picks a sequence of 2 coin tosses (i.e. one of HH / TT / HT / TH)
Step (2)
Find a biased coin which has 2/3's chance of landing on H and 1/3 chance of landing on T
Step (3)
Flip! Repeatedly.
We flip the biased coin until one of us sees our sequence. Whoever sees it first, wins.
For instance, I pick TH, you pick HT, and our flips go T T H, so I win.
Query:
Flashback to step one: Do you want to pick the initial sequence first or second if you wish to win this game?
GL!
edit:tidy, puzzle is the same
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+ Show Spoiler +I pick first so I can pick HH, right? Because H is the most likely one to come at each flip and thus the sequence of HH should be the most likely as well..
Disclaimer: I might be an idiot or missing something crucial.
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Solution should be something like:
+ Show Spoiler +You want to choose second. Reason as follows:
If the person who picks first takes HH then in response you take TH. If a tails occurs at any point you have to win (string of tails until one heads at which point the last two coins are TH and you win). Thus the only way you can lose is if the first two coins are HH. Probability of this is 4/9, so you win with probability 5/9.
Opponent picks TH, you pick HT. Since if the first coin is a heads you win, a tails you lose. Thus winning probability is 2/3.
Opponent picks HT. You pick HH. Winning probability is 2/3.
Opponent picks TT. Opponent is pretty crazy, you're more likely to win whatever you pick.
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i would pick HH as well...
chances are 4/9...
i don't know...it's confusing fuck
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On March 04 2010 05:43 Puosu wrote:+ Show Spoiler +I pick first so I can pick HH, right? Because H is the most likely one to come at each flip and thus the sequence of HH should be the most likely as well.. Disclaimer: I might be an idiot or missing something crucial.
+ Show Spoiler +stats are fun eh? intuition is almost never the way to go XD
this wasn't a hard puzzle, i just thought it was interesting -- see the 2nd reply for details
to clarify here though: HH has a 2/3*2/3 = 4/9 chance of winning right away. However, TH wins in every other case which occurs 5/9 of the time.
If you get any of the other cases, TH is going to appear before HH
that's just this example, the 2nd reply in the thread goes into detail on all 4.
Having said that, try and work out the other 3 cases for yourself, the logic is pretty similar, and it's fun xD
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South Africa4316 Posts
+ Show Spoiler +You want to choose second, because you can always choose something that has at least a 33% chance of happening before his sequence starts, and a 33% of 66% (eg. a 22%) chance of happening after his sequence starts, giving you a 55% chance of winning).
For example, if he chooses HH, you choose TH. 33% of the time tail will occur first, and you will definitely win. 66% of the time, head will occur first, meaning you have a 33% chance of still winning.
If he chooses HT, then you choose HH. That gives you a 66% chance that an H will follow an H rather than a T following an H.
If he chooses TH then you choose HT, giving you a 66% chance hitting heads directly and winning.
If he chooses TT, you choose HT, giving you a 66% chance of winning straight up (if it starts H), and another 66% chance of winning if the 33% tail starts.
All in all, you always have a better chance of winning if you can choose first.
Is this right?
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Not sure if I should spoiler this, it's just a small observation, and it's likely that the puzzle has already been solved.
+ Show Spoiler +A HH first pick is countered by TH second pick. The person who picked TH has (Tails first) + (Head first tails second) = 1/3 + 2/3*1/3 = 5/9 chance to win. Edit: I made calculations. + Show Spoiler +
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EPT![[image loading]](http://i45.tinypic.com/24l7zlu.jpg)
