Pennies  challenge
The Game is not fair !
Here is why:
Player A needs Head followed by Tail or HT to win.
Player B needs Head followed by Head or HH to win.
Let's examine the results after K trials. (2**K total possibilities).
What is the number of possibilities for Player A NOT to win.
versus the number of possibilites of Player B NOT to win.
Case 1: no heads at all
Player A and B have one such possibility
Case 2: 1 head found
Player A and B have the same K possibilities.
Case 3: 2 heads found
Player A must have the Heads at the end
which is a single possibility. and it looks like:
T T T T ..... T T H H
Player B has much more
For him not to win he need the construct .... H T H but
also ... H T T H in fact in this case he as
(K1)*(K2)/2 possibilites
In fact for the case of 3 heads Player has one possibility
over the K trial ..... T T T H H H
however Player B has number of possibilities in the order
of O(3) or K**3 compared to Player A.
And so on and so forth ...
Therefore clearly Player A has an advantage.
Aharon Tam
Wednesday, July 16, 2003
I think the correct solution is that there is no advantage to either player. The two published solutions seems to miss that the game ends as soon as one player wins.
A needs HT, B needs HH
2 Tosses:
HH B Wins
HT A Wins
TH No winner
TT No winner
A and B have same odds of winning. Only TH and TT play on.
3rd toss:
THH B Wins
THT A Wins
TTH No winner
TTT No winner
A and B have same odds of winning. Only TTH and TTT play on.
4th toss:
TTHH B Wins
TTHT A Wins
TTTH No winner
TTTT No winner
A and B have same odds of winning. Only TTTH and TTTT play on.
5th toss:
TTTHH
TTTHT
TTTTH
TTTTT
etc, etc.
Lars
Thursday, September 18, 2003
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