What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? Or other useful resources. 



> 
> We can complicate matters by allowing transition probabilities
>  to vary from 1/2 and by allowing steps longer than one
>  unit. Consider the following curious paradox first called to my 
>  attention (in betting terms) by Enn Norak, a Canadian mathematician.
>  A walker starts 100 steps to the right of 0 on a line
>  that has no barriers. Instead of a coin a packet
>  of 10 playing cards-five red and five black-is used as a randomizer.
>  The cards are shuffled and spread face down and any
>  card is selected. After its color is noted it is discarded. If it is red, the walker steps to the right: if black, he steps to the left.
>  This continues until all 10 cards have been taken. (The transition
>  probability varies with each step. It is 1/2 only when there
>  is an equal mixture of red and black cards before the draw.)
>  The walk differs also from walks discussed above in that before
>  each card is noted the walker chooses the length (which need
>  not be integral) of his next step.
>  Assume that the walker adopts the following halving strategy
>  in choosing step lengths. After each card is noted he takes a step
>  (left or right) equal to exactly half of his distance from 0. His
>  first step is 100/2 = 50 units. If the card is red, he goes to the
>  150 mark. His next step will then be 150/2 = 75. If the first
>  card drawn is black, he goes left to the 50 mark, and so his next
>  step will be 50/2 = 25. He continues in this manner until the
>  tenth card is noted. Will he then be to the right or to the left of
>  the 100 mark where he began the walk?
>  The answer is that he is sure to be to the left. This may not
>  be very surprising, but it is surely astonishing that, regardless 
>  of the order in which the cards are drawn, he will end the walk at 
>  exactly the same spot.
> 
> 
> 



Puzzle in Martin Gardner book
Sure, multiplication is commutative, but there is more to it than that. While being reasonably easy, this puzzle suggests variations in ways that the equation $x\cdot y = y\cdot x$ doesn't. 


In his wonderful paper Games People Don't Play, <http://www.teorver.ru/newkatalog/1193689162.pdf>, Peter Winkler describes essentially the same game as ``Next card color betting'' (a bit of googling also turns up <http://www.maxdama.com/?p=137> and <http://www.dartblog.com/data/2008/08/007950.php>). But there the player, Victor, wants to end up as far to the right as possible (increasing his bankroll). It turns out that there is a strategy that guarantees him to end up with $2^{10}/\binom{10}{5} = 256/63$ times his initial bankroll, or about 406 steps to the right of the origin.


This is a game that children can understand, but if we pursue the analysis, it doesn't stop until we have developed, besides insights into hedging strategies, a good deal of nontrivial mathematics including information theory (the amount of information Victor has about the red-black sequence dictates exactly how much money he will ideally make by betting), the Wallis product formula (showing that his final bankroll is asymptotically $\sqrt{\pi n}$ for a deck of $n$ red and $n$ black cards), and even the central limit theorem. 


