Wow, what an amazing statement. Any time someone says to me
"in mathematics, there is only one correct way" I can't help but
conclude that the person hasn't been exposed to very much math.
Example: in geometry, problems can be solved using Cartesian
coordinates or with polar coordinates or cylindrical coordinates.
Different ways to view the same problem.
Here things are slightly different, in that the difference comes in
how one chooses to define which events mark the beginning
and end of a "game". There are plenty of examples of games
where the boundaries can be defined in different ways, and the
choice alters the resulting numbers. Of course, this makes the
comparison of EV an apples to oranges comparison, so players
who use different definitions of "game" can't expect to be able
to compare numbers directly.
Here's another example where two different players view the
same game differently: In Craps, a $6 place bet on number 8
is paid $7 if an 8 is rolled before the next 7. These bets can
be placed or removed at any time. Jim defines the end of a
game as a roll of 8 or 7, where the 8 results in a $7 win and
a 7 results in a $6 loss. On average a seven is rolled 6 times
for every 5 times that the roll is eight. Jim computes his EV
as 100 * [($7 * 5) - ($6 * 6)] / ($6 * 11) = -1.515%.
Bob defines each roll as a complete game. In fact, if the roll
is neither an 8 or a 7, he superstitiously removes the $6 wager
and replaces it with a new $6 wager using different chips (he
hates to bet the same chips twice in a row). This is perfectly legal
-- the casino allows place bets to be changed or removed at any
time. So, Bob sees his probability of winning as 5/36, his
probability of losing as 6/36, and his probability of a "push"
as 25/36. From Bob's perspective, the EV of the game is
[($7 * 5) + ($0 * 25) - ($6 * 6)] / ($6 * 36) = -0.4630%
Both of these ways of viewing the games are mathematically
correct. On average, Jim plays only 11 games in the same
amount of time that Bob plays 36 games. If they bet on the
same Craps table and make equal wagers and always bet
on the same rolls, then they are guaranteed to win or lose
identical sums of money. The only difference is how they
perceive the rolls that are neither 7 nor 8. Bob perceives
them as a "push" while Jim perceives them as a non-event.
Two very different EVs from defining the same game in two
very different ways.
···
On Wednesday 31 March 2004 05:52 pm, Dan wrote:
Steve Jacobs wrote (snip):
> > >If, on the other hand, you consider this to be a single event, you
>> >will consider it to be a return of 10 coins with a coin-in of 5.
>>
>> Exactly right. Twice the payoff with half the probability of receiving
>> it.
>
>That is one correct way of viewing it, and nobody is claiming that
>your way of viewing it is wrong. It just isn't the _only_ way.
I suggest you find a statistician who is familiar with game theory
and ask about that. In mathematics, there is only one correct way.