TO FLIP OR NOT TO FLIP

Hey!

I've been reading a lot of evaluations of Pokémon cards on POJO, and a
recurring error in these reviews concerns the probability of getting a
certain result when flipping coins.

OK, cool it.  I'm not talking about attacks or Pokémon Powers that involve
flipping one coin, I'm talking about attacks that involve flipping many coins
(Cool Porygon's 3D Attack, Jolteon's Pin Missile, Kangaskhan's Comet Punch,
etc.).

Right then, as most people think that maths is hard, we shall consider an
EXAMPLE before moving onto the GENERAL RULE.  I don't know if this maths is
hard or not (I am English-educated and don't know the US school system), but
copying the example below should allow you to do this - and you can always
print this out (it is in ASCII format after all) and ask your maths teacher
at school for help!

Consider Cool Porgon's 3D Attack ("flip 3 coins, this attack does 20x the
number of heads"); some reviews I've read state that there is a 25% chance of
doing 0, 20, 40, and 60 damage, but this is wrong.  The easiest way to figure
out the actual chance of doing a certain amount of damage with the 3D Attack
is to look at all the possible coin combinations.

Consider an unbiased coin (a coin that has an equal chance of flipping heads
or tails), and let H = heads and T = tails.  The combinations are:

    TTT
    HTT
    THT
    TTH
    HHT
    THH
    HTH
    HHH

There are a total of eight combinations, hence, there is a 1/8 chance of
doing no damage, a 3/8 chance of doing 20 damage, a 3/8 chance of doing 40
damage, and a 1/8 chance of doing 60 damage.

The main error that seems to be made when determining the chance of doing a
certain amount of damage is that only the number of OUTCOMES are considered,
not the actual COMBINATIONS that lead to the outcomes.  For all cases other
than flipping one coin, there are always more COMBINATIONS than OUTCOMES.

OK, now the GENERAL CASE.  This method will allow you to determine the number
of combinations and assign the chance of getting a certain outcome for any
number of coin flips (usually limited in the Pokémon TCG to 4 coin flips,
although Exeggutor offers an unlimited number ;)).

For x number of coins (where x is any number), the number of combinations of
heads and tails is given by 2 raised to the power x.  For all those not
familiar with algebra, this translates as:

1 coin  = 2 x 1 combinations        = 2  combinations
2 coins = 2 x 2 combinations        = 4  combinations
3 coins = 2 x 2 x 2 combinations    = 8  combinations
4 coins = 2 x 2 x 2 x 2 combinations    = 16 combinations

The probability of a certain number of heads can be determined by reference
to the xth line of Pascal's Triangle:

1   1,1     2
2   1,2,1       4
3   1,3,3,1     8
4   1,4,6,4,1   16
5   1,5,10,10,5,1   32

Hence, for an attack that flips 4 coins, the chance of getting a certain
number of heads is determined by reference to the 4th line of Pascal's
triangle, and the probabilities are assigned left to right:

0 heads = 1/16
1 Head  = 4/16  = 1/4
2 Heads = 6/16
3 heads = 4/16  = 1/4
4 Heads = 1/16

If you followed that first time, well you are fairly smart.  If you didn't,
Consider the simpler case of 2 coins where there are four combinations, and
the chance of getting a certain number of heads is determined by reference to
the 2nd line of Pascal's triangle:

0 heads = 1/4
1 Head  = 2/4   = 1/2
2 Heads = 1/4

If you are still not sure, the analysis presented here can be summarised by
the following main points:

-   The chance of getting all heads is the same as the chance of getting no
heads.
-   The most probable number of heads lies midway between the minimum and
maximum number of   heads.
-   The more coins that are flipped, the greater the chance of flipping the
most probable       number of heads.

This article has probably created more confusion than there was before, but
hey, I've tried my best.  I live in the UK so I'm probably in bed when you
are on-line, but you can e-mail me at

philipbinns@AOL.com

Thanks for reading the whole article everyone!  This was my first article!

Phil

If this article is published on POJO, It might inspire me to write other
articles - probably about the Pokémon TCG and NOT higher mathematics.  I'll
flip a coin to help me decide...