Two people, call them $A$ and $B$, decide to play a card game. They take 2 standard decks of playing cards and combine them into a "superdeck" of 104 cards, shuffle them well, and then draw 1 card at a time randomly without replacement. $A$ immediately wins if 13 cards of any single suit are drawn. $B$ immediately wins if at least 18 black and at least 18 red cards are drawn. If they both "win" on the same card draw it is a tie (no decision) and they start over with all 104 cards after reshuffling. They bet even money—dollar for dollar matching. Who, if anyone, has the advantage of winning and by how much?
Note that all cards are shared "community" cards so the players are not drawing their own set of cards.
If someone would like to comment on if and how the probability changes if they draw their cards in tandem, that would be very interesting and informative. For example, two cards would be drawn at a time, giving one to $A$ and one to $B$, and then checked if anyone won or if they tied. This is an optional bonus question and not required to earn the checkmark for best answer.
Simulating 1,000,000,000 (1 billion) decisions (including ties) of the original question (shared cards), I got the following winning percentages: $$ A\quad 38.7855918\%\\ B\quad 59.7770491\%\\ \textrm{Tie}\quad 1.4373591\% $$
Minimum # of cards : $14$ (A disappointment as I was expecting it to be $13$)
Maximum # of cards : $42$
Average # of cards : $37.1902$
Note that at "first glance", some people might think that $A$ has the advantage because $A$ can win with as few as 13 cards but B needs 36 cards at a minimum to win. However, look at how much $B$ is actually favored to win. Another thing that somewhat surprised me is the average number of cards for someone to win (or tie) is about 37, only about 1 card more than $B$'s minimum needed to win. That might also imply that $A$ has an advantage. What if I reworded the question to also state that the average number of cards for a decision is slightly over 37, would most people then think at first glance that $A$ has an advantage, possibly a huge one? Yet another thing that might mislead people into thinking that $A$ has a large advantage is that $A$ is guaranteed not to lose if at least 42 cards are drawn. At that point either $A$ has won or tied. Kudos to the person that suspected that $B$ was the favorite from the start because he didn't let these false biases sway him towards $A$ being the favorite.
UPDATE:
I ran the alternate simulation giving each player their own cards and got some interesting results. $B$ remains the favorite at about $57.1\%$ to $A$'s $36.8\%$ which means the ties more than quadrupled to about $6.1\%$. The average number of cards slightly more than doubled from about 37 to about 75 with a low of 28 and a high of 94 (in ten million decisions).
Perhaps one way to simplify getting the answer to this original problem is to start the game by immediately turning over 35 cards since $B$ cannot win with less than 36 cards. Check if $A$ won at 35 cards. If not, then draw cards 36 thru 42 one at a time checking for a winner. By draw 42 someone will have won or there is a tie. So would the prob of $A$ winning on a 35 card draw be: $$ \large \frac{4\times {26 \choose 13} \times {78 \choose 22}}{{104 \choose 35}} $$
That is about $10\%$. There may be two 13 card flushes in that but we don't care as long as there is at least 1.
Also it is interesting to note that in the original game with shared cards, $42$ cards is the max for a win or tie because of the $25+17$ situation but with separate hands, that doesn't happen so the max # of cards drawn can exceed $84$ (which is $2$ * $42$) and in my simulation is did exceed it as it maxxed out at $94$ cards, not $84$. For example, B could get $29$ black cards and $17$ red cards but A could have $12, 12, 11,$ and $11$ of each suit (respectively) so at that point, neither is a winner and $2$ more cards would be drawn (almost exhausting the deck of $104$). The max number of cards ever for the separate card variation of this game seems like $94$ but I am not certain yet.
Another interesting point is that simulation of the separate card variation is considerably slower because the average approximate number of cards needed is about double ($75$ vs. $37$) and can go as high as $94$ which is almost the entire deck so it becomes "harder" (slower).