In a 2-player game of Texas Hold 'em poker, player vs dealer, the dealer shuffles the deck of cards and deals himself $7\clubsuit,7\spadesuit$ and the player $6\heartsuit,6\diamondsuit$. He then draws the flop. What is the probability that these 3 cards will give the player a better chance statistically of winning the game? My solution so far: There are $\binom {48}3 = 17,296$ possible combinations of these three cards.This will be the denominator to find the probability but I am stuck on how to find the $3$ cards that will give the player a $>50\%$ chance of winning after the flop has been drawn.

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    $\begingroup$ I've rolled back - do not deface your question. $\endgroup$ – Noah Schweber Sep 13 '18 at 21:27
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    $\begingroup$ Also do not change your question to something completely unrelated - if you have another question, ask another question. $\endgroup$ – Noah Schweber Sep 13 '18 at 21:31
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    $\begingroup$ Daten: this is the first of two questions you've tried to render illegible, after having received an answer. That makes some of us wonder if you are trying to hide your on-line footsteps. If using this site for homework, or take home test questions, or online questions is expressly forbidden by the professors or institutions they teach at, know that you can't "game this site that way". You ask a question and you get an answer = your question stays put, unless after first being closed, it is then deleted. (Attempts to delete posts can lead to a ban on asking questions on this site.) $\endgroup$ – Namaste Sep 13 '18 at 21:39
  • $\begingroup$ @amWhy You have more reputation than me - can you lock questions, or does that happen further up the hierarchy? (If you can, this one might be worth locking.) $\endgroup$ – Noah Schweber Sep 13 '18 at 21:40
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    $\begingroup$ No, only mods can "lock" a post, or under some combination of circumstances (number of flags, what have you, "Community diamond" will intervene to lock. We can "protect" a question, but that means very little in terms of preventing further answers andor further defacing of a question by a user. Best to flag a mod. $\endgroup$ – Namaste Sep 13 '18 at 21:43

In order for the player to win, he must hit a 3 of a kind or a straight or a flush.

3 of a kind: What is the chance a 6 falls in the flop?

And other contingencies to consider... yes, the dealer could also flop a 3rd 7.

We need to see a 6 but no 7. Or 6-6-7 on the flop.

Straight or flush: The player can do no better than 4 cards toward a straight or a flush. If he hits this on the flop, what is the chance that he converts with the remaining two cards? If it is less than 50% you can dismiss these cases.


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