When we play out a split hand - we have the follwing 2 possible types of second card that can be

played onto a split card - either the paircard (P) or a non-pair card (N).

Since the number of hands that can result are limited by the rules, there are only two possible

hands that can be end points of a split card. Either a non-split card was dealt (N) - or the

maximum number of hands has be reached and any card including a paircard can be played (x) as the

second card of the hand. These hands do not need to consist of only 2 cards, however N and x refer

to which second cards are allowed. Note that "N" can refer to either the non-paircard or to the

hand which began with a non-paircard depending on the context.

A round for the player can consist of any of the possible different hands that will be played

until either the maximum number of splits is reached or until no free pair cards are left.

The probability of the hands and therefore the round is determined by the N and P cards played,

while the EV of the hands/round is determined by the type of hands that result, i.e. N and x.

The p(N) is simply 1-p(P). Determining the EV's of N and x hands is the hard part.

In the case of infinite splits, p(N), p(x), EV(N) and EV(x) are not dependent on previous hands

and therefore do not change. An additional notation will be introduced for finite splits to

account for the effects of removal on N and x hands. For reference, the different possible rounds

that can occur as determined by the maximum number of splits allowed is listed below:

Hand EV Prob

SPL1

xx 2*EV(x) 1

SPL2

NN 2*EV(N) p(N)*p(N)

Pxxx 3*EV(x) p(P)

NPxx EV(N)+2*EV(x) p(N)*p(P)

SPL3

NN 2*EV(N) p(N)*p(N)

PNNN 3*EV(N) p(P)*p(N)*p(N)*p(N)

NPNN 3*EV(N) p(N)*p(P)*p(N)*p(N)

PPxxxx 4*EV(x) p(P)*p(P)

PNPxxx EV(N)+3*EV(x) p(P)*p(N)*p(P)

NPPxxx EV(N)+3*EV(x) p(N)*p(P)*p(P)

PNNPxx 2*EV(N)+2*EV(x) p(P)*p(N)*p(N)*p(P)

NPNPxx 2*EV(N)+2*EV(x) p(N)*p(P)*p(N)*p(P)

SPL4

NN 2*EV(N) p(N)*p(N)

PNNN 3*EV(N) p(P)*p(N)*p(N)*p(N)

NPNN 3*EV(N) p(N)*p(P)*p(N)*p(N)

PPNNNN 4*EV(N) p(P)*p(P)*p(N)*p(N)*p(N)*p(N)

PNPNNN 4*EV(N) p(P)*p(N)*p(P)*p(N)*p(N)*p(N)

PNNPNN 4*EV(N) p(P)*p(N)*p(N)*p(P)*p(N)*p(N)

NPPNNN 4*EV(N) p(N)*p(P)*p(P)*p(N)*p(N)*p(N)

NPNPNN 4*EV(N) p(N)*p(P)*p(N)*p(P)*p(N)*p(N)

PPPxxxxx 5*EV(x) p(P)*p(P)*p(P)

PPNPxxxx EV(N)+4*EV(x) p(P)*p(P)*p(N)*p(P)

PNPPxxxx EV(N)+4*EV(x) p(P)*p(N)*p(P)*p(P)

NPPPxxxx EV(N)+4*EV(x) p(N)*p(P)*p(P)*p(P)

PPNNPxxx 2*EV(N)+3*EV(x) p(P)*p(P)*p(N)*p(N)*p(P)

PNPNPxxx 2*EV(N)+3*EV(x) p(P)*p(N)*p(P)*p(N)*p(P)

PNNPPxxx 2*EV(N)+3*EV(x) p(P)*p(N)*p(N)*p(P)*p(P)

NPPNPxxx 2*EV(N)+3*EV(x) p(N)*p(P)*p(P)*p(N)*p(P)

NPNPPxxx 2*EV(N)+3*EV(x) p(N)*p(P)*p(N)*p(P)*p(P)

PPNNNPxx 3*EV(N)+2*EV(x) p(P)*p(P)*p(N)*p(N)*p(N)*p(P)

PNPNNPxx 3*EV(N)+2*EV(x) p(P)*p(N)*p(P)*p(N)*p(N)*p(P)

PNNPNPxx 3*EV(N)+2*EV(x) p(P)*p(N)*p(N)*p(P)*p(N)*p(P)

NPPNNPxx 3*EV(N)+2*EV(x) p(N)*p(P)*p(P)*p(N)*p(N)*p(P)

NPNPNPxx 3*EV(N)+2*EV(x) p(N)*p(P)*p(N)*p(P)*p(N)*p(P)

SPL5-Infinity - Enjoy