Pennies So what am I missing here? The solution to "Pennies":
Joe Fordham
Which part of "we each get one penny" didn't you understand?
Skeptic
So X is me, wanting heads-then-tails. O is you, wanting heads-then-heads. For shorthand, (H|T) means a simultaneous flip where I flipped heads and you flipped tails.
Tail of the "g"
These can happen where O wins.
JHY
Yep, definitely jinxed it. I completely forgot to draw the edges from each node to itself on my state diagram. Grumble.
Tail of the "g"
O looks for HH
WanFactory
I calculated the average number of rounds before each person won: the result was 4 for the heads then tails person and 6 for the two heads in a row person.
John
Ok, I think I found a full solution: probability of the heads-then-tails person (call that X, the other O) winning is 5/8 (assuming that they start over replay on a tie).
WanFactory
All the above "answers" miss a key point. The game continues till someone wins. That means you can't enumerate all the outcomes. Te guy who wants HH can flip an arbitrarily large number of tails. The other guy simultaneously flips the same arbitrarily large number of heads. Then one of them gets lucky.
Alex. McLellan
Making the same point, but more specifically...
Alex. McLellan
I believe my answer takes into the account the "continue" situations.
WanFactory
H:H H:T T:H T:T
Mike
Great, let's play for even money. I'm looking for heads-then-tails. You look for heads-then-heads.
WanFactory
No, I mark H:H --> H:T as a draw. You are confusing conditional probability with a joint probability of two independent events. Each cell in the table has an equal probabolity of 1/16.
Mike
In the position when X and O both have tails, none of them has a chance to win at the next toss, but when both have heads - a probability to win for both is 1/4 ( not 1/2 'cause the draw is also a possibility). The rest is symmetric.
Mike
Lets play for money
WanFactory
Let me try this again:
WanFactory
Now I got it. The game is not reset. The "DROW" still contains a possibility for you and no chance for me. This is where I am being bobbed. Simple simulation confirms you are right . Thanks.
Mike
Unfortunately, I'm not sure if my 5/8 solution is correct. I simply do not see any flaws in my reasoning - which does not mean very much.
WanFactory
Here's a little clearer an argument for those who are still having trouble following (like I was a moment ago). I'm pretty sure my math is right.
Krease
With each flip of the penny the number of head/tail combinations doubles. There are only 4 combinations after 2 flips, but 8 after 3. In general after n flips, there are 2^n combinations. Many of these combinations will not be applicable because there was a previous match (HT for player X, and HH for player O) in its history. The question then is: how many of these combinations are new matches?
slava
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