$2$ people (C and D) decide to play a computer assisted game. The computer is programmed to quickly play as many fair hands (using the equivalent of a fair $52$ card deck), until someone wins the hand so the win will appear instantly when they press a button on the keyboard and then it waits for the next keypress for the next game. The computer is necessary because of how infrequent wins are, thus a real deck of cards would take way too long between wins.
The rules are player C can win if he gets $3$ "gapped" $3$ card straights dealt in order. An A (ace) will be considered the highest rank for both C and D so $A23~567~9TJ$ (T = ten rank) would NOT be a win for C but $234~QKA~789$ would be. There has to be at least a gap of $1$ rank between the $3$ card straights but can be wider as shown in the 2nd example. Note that the individual straights of length $3$ have to be dealt in order but the actual $3$ length $3$ straights don't need to be in relative order (as shown by $234QKA789$). That is still a win for C and need not be $234789QKA$ (although that would also be a win for C). The simplest example of a win for C would be something like $234678TJQ$. This is also the lowest ranked win for C. The highest rank win for C would be $45689TQKA$. Remember permutations are allowed such as $89TQKA456$ is also a win but $243678TJQ$ is not. If you think of each gapped $3$ card straight as a letter. Such as with $234678TJQ$, X=$234$, Y=$678$, Z=$TJQ$, then valid permutations are XYZ, XZY, YXZ, YZX, ZXY, and ZYX.
For player D to win, $2$ pattern classes are allowed. Either any $6$ card straight (such as $345678$) or this pattern: $223344$, $334455$, $556677$... $QQKKAA$.
The computer will keep dealing community (shared) cards until either there is a winner of the hand (not likely) or all $52$ cards are depleted (very likely). The cards are reshuffled after each win and after each nonwin (all $52$ cards are depleted). Permutations are NOT allowed. The straights and patterns MUST be dealt in order. For example, $345678$ is a win for D but $357468$ is not, even though they could be the same exact cards. Suits are irrelevant for the straights and other winning patterns. Any suits are allowed.
So it is a rainy day and they play many hundreds of winning hands to see who gets more wins. The question is who has the mathematical advantage and by how much? A hand that ties can be ignored as they will not count that hand and will continue play. Neither player has any knowledge of the expected probability and both just play on a "hunch" based on the observed patterns to win which they are told about.
The $300$ bounty ends Monday night (Aug 22nd) around 11:59PM (Eastern time) already including the $24$ hour grace period. Other simulations are encouraged but also math solutions or partial math solutions (like how to set it up properly).
Sample data of $3$ million random shuffles can be downloaded here for test purposes and/or analysis for mathematical solutions.:
Format is $52$ bytes of data per line and $2$ more bytes for the end of line so $54$ bytes total per line, $3$ million lines. "$2$" = rank $2$, "$3$" = rank $3$... "T" = rank $10$.... All cards are represented by this set of ranks: "$23456789TJQKA$". There are $4$ of each rank in each shuffle (suits are irrelevant in this card game).
Where are the mathematical attempts at solving the probabilities of this card game? How about even some partial solutions like take the winning D patterns and compute the probability of a D win if D was the only player, subtracting out the overcounting cases where $2$ (or more) of those patterns can appear in the same shuffled deck. For example, if we get something like $234567223344$