Cards - Revision history

Admin: /* Straight */

2019-03-09T21:42:16Z

Straight

← Older revision		Revision as of 21:42, 9 March 2019
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	</math>		</math>

	This gives the desired count of hands ~~winning~~ with a straight:		This gives the desired count of hands with a straight:

	<math>		<math>
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	Player 1 got straight 3625 / 1000000 = 0.00362500		Player 1 got straight 3625 / 1000000 = 0.00362500
	</pre>		</pre>


	===Three of a Kind===		===Three of a Kind===

Admin: /* Straight Flush */

2019-03-09T21:40:27Z

Straight Flush

← Older revision		Revision as of 21:40, 9 March 2019
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	</math>		</math>

	Notice that the use of the Principle of Inclusion/Exclusion was ~~straightforward and~~ somewhat subtle ~~here~~. As we accumulate a wider variety of hands, we will need more complicated inclusion/exclusion counts.		Notice that the use of the Principle of Inclusion/Exclusion here, while straightforward, was also somewhat subtle. As we accumulate a wider variety of hands, we will need more complicated inclusion/exclusion counts.

	Here is output from the poker hand counting program:		Here is output from the Monte Carlo poker hand counting program:

	<pre>		<pre>

Admin: /* Straight Flush */

2019-03-09T21:38:10Z

Straight Flush

← Older revision		Revision as of 21:38, 9 March 2019
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	Using the principle of inclusion-exclusion, we can write the number of hands with a straight flush outcome as:		Using the principle of inclusion-exclusion, we can write the number of hands with a straight flush outcome as:

	~~(Hands with~~ straight flush) = ~~(Hands with~~ royal flush) + ~~(Hands without royal~~ flush)		<math>
			C_{\mbox{straight flush}} = C_{\mbox{royal straight flush}} + C_{\mbox{unroyal straight flush}}
			</math>

	~~Then we can~~ exclude ~~the~~ hands with a royal flush, since we already counted them in the royal flush outcomes above.		We should exclude hands with a royal flush, since we already counted them in the royal flush outcomes above. If we included them, we would be over-counting straight flushes.

	The procedure for a straight flush is similar to the procedure for a royal flush: we begin by selecting a suit, which is <math>\binom{4}{1}</math>, then we go on down the line and select our cards. If a straight consists of 5 cards, the straight must start with a card between 2 and 10. If the straight starts with 10, it is a royal flush and we have already counted it. Therefore, the first card in the straight flush can be any one of the cards A through 9. That's 9 total choices of cards, from which we are selecting one, for <math>\binom{9}{1}</math>.		The procedure for a straight flush is similar to the procedure for a royal flush: we begin by selecting a suit, which is <math>\binom{4}{1}</math>, then we go on down the line and select our cards. If a straight consists of 5 cards, the straight must start with a card between 2 and 10. If the straight starts with 10, it is a royal flush and we have already counted it. Therefore, the first card in the straight flush can be any one of the cards A through 9. That's 9 total choices of cards, from which we are selecting one, for <math>\binom{9}{1}</math>.

Admin at 21:34, 9 March 2019

2019-03-09T21:34:29Z

← Older revision		Revision as of 21:34, 9 March 2019
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	Mathematics related to cards.		Mathematics related to cards.

	This relies heavily on [[AOCP/Binomial Coefficients]] and the [[Hypergeometric Distribution]] (~~which describes~~ sampling without replacement).		This relies heavily on [[AOCP/Binomial Coefficients]] and the [[Hypergeometric Distribution]] (counts for sampling without replacement).

	All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards		All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards

Admin at 09:04, 9 March 2019

2019-03-09T09:04:05Z

← Older revision		Revision as of 09:04, 9 March 2019
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	Mathematics related to cards.		Mathematics related to cards.

	This relies heavily on [[AOCP/Binomial Coefficients]] and the [[Hypergeometric Distribution]] (which describes sampling ~~with~~ replacement).		This relies heavily on [[AOCP/Binomial Coefficients]] and the [[Hypergeometric Distribution]] (which describes sampling without replacement).

	All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards		All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards

Admin at 09:03, 9 March 2019

2019-03-09T09:03:45Z

← Older revision		Revision as of 09:03, 9 March 2019
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	Mathematics related to cards.		Mathematics related to cards.

	This relies heavily on [[AOCP/Binomial ~~Coefficient~~]] and the [[Hypergeometric Distribution]] (which describes sampling with replacement).		This relies heavily on [[AOCP/Binomial Coefficients]] and the [[Hypergeometric Distribution]] (which describes sampling with replacement).

	All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards		All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards

Admin at 08:55, 9 March 2019

2019-03-09T08:55:38Z

← Older revision		Revision as of 08:55, 9 March 2019
Line 1:		Line 1:
	Mathematics related to cards.		Mathematics related to cards.

			This relies heavily on [[AOCP/Binomial Coefficient]] and the [[Hypergeometric Distribution]] (which describes sampling with replacement).

	All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards		All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards

Admin: /* The Abbreviated Table */

2019-03-09T08:48:28Z

The Abbreviated Table

← Older revision		Revision as of 08:48, 9 March 2019
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	\|Royal Flush		\|Royal Flush
	\|1 / 649,740		\|1 / 649,740
	\|~1.~~5391e-6~~		\|~0.0000015391

	\|-		\|-
	\|Straight Flush		\|Straight Flush
	\|3 / 216,580		\|3 / 216,580
	\|~1.~~385e-5~~		\|~0.00001385

	\|-		\|-
	\|Four of a Kind		\|Four of a Kind
	\|1 / 4165		\|1 / 4165
	\| ~2.~~40e-4~~		\|~0.000240

	\|-		\|-

Admin: /* The Abbreviated Table */

2019-03-09T08:47:00Z

The Abbreviated Table

← Older revision		Revision as of 08:47, 9 March 2019
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	\|-		\|-
	\|Royal Flush		\|Royal Flush
	\|		\|1 / 649,740
	\|		\|~1.5391e-6

	\|-		\|-
	\|Straight Flush		\|Straight Flush
	\|		\|3 / 216,580
	\|		\|~1.385e-5

	\|-		\|-
	\|Four of a Kind		\|Four of a Kind
	\|		\|1 / 4165
	\|		\| ~2.40e-4

	\|-		\|-
	\|Full House		\|Full House
	\|		\|6 / 4165
	\|		\|~0.001440

	\|-		\|-
	\|Flush		\|Flush
	\|		\|12,777 / 649,740
	\|		\| ~0.0019

	\|-		\|-
	\|Straight		\|Straight
	\|		\|5 / 1,274
	\|		\| ~0.003924

	\|-		\|-
	\|Three of a Kind		\|Three of a Kind
	\|		\|88 / 4,165
	\|		\|~0.0211

	\|-		\|-
	\|Two Pairs		\|Two Pairs
	\|		\|198 / 4,165
	\|		\|~0.04753

	\|-		\|-
	\|One Pair		\|One Pair
	\|		\|353 / 833
	\|		\|~0.422569

	\|}		\|}

Admin at 04:56, 22 February 2019

2019-02-22T04:56:40Z

← Older revision		Revision as of 04:56, 22 February 2019
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	Mathematics related to cards.		Mathematics related to cards.

			All code referenced on this page is available here: https://git.charlesreid1.com/cs/java/src/branch/master/combinatorics/cards

	=Combinatorics and Counting=		=Combinatorics and Counting=