<HTML>਀㰀吀䤀吀䰀䔀㸀倀漀欀攀爀 伀搀搀猀㰀⼀吀䤀吀䰀䔀㸀 
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਀㰀℀ⴀⴀऀ㰀䄀 䠀刀䔀䘀㴀∀樀愀瘀愀猀挀爀椀瀀琀㨀伀瀀攀渀圀椀渀⠀✀唀刀䰀✀⤀∀㸀㰀䈀㸀一䄀䴀䔀㰀⼀䈀㸀㰀⼀䄀㸀 ⴀⴀ㸀 
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਀㰀䈀伀䐀夀 䈀䜀䌀伀䰀伀刀㴀∀⌀䐀䐀䐀䐀䐀䐀∀㸀 
<TABLE BORDER="1" BGCOLOR="white" CELLPADDING="5" WIDTH="100%"><TD>਀ऀ㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀∀㈀∀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀∀㔀∀ 䈀䜀䌀伀䰀伀刀㴀∀⌀䄀䄀䌀䌀䄀䄀∀㸀㰀吀䐀㸀 
	<FONT SIZE="+2"><B>Odds & Probabilities in Texas Hold'em</B></FONT>਀ऀ㰀⼀吀䄀䈀䰀䔀㸀 
<FONT SIZE="-2">motivated by e-mail from  <A HREF="javascript:OpenWin('http://www.pokercalculatorreport.com/')"><B>Marty S.</B></A></FONT>਀㰀䈀刀㸀㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀㈀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀㔀㸀㰀吀䐀㸀 
In Texas Hold'em, you're dealt two cards.਀㰀䈀刀㸀匀甀戀猀攀焀甀攀渀琀氀礀Ⰰ 琀栀攀 搀攀愀氀攀爀 瀀氀愀挀攀猀 琀栀爀攀攀 挀愀爀搀猀 昀愀挀攀 甀瀀 漀渀 琀栀攀 琀愀戀氀攀㨀 琀栀攀 㰀䈀㸀䘀氀漀瀀㰀⼀䈀㸀⸀ 
<BR>Then he places another face up: the <B>Turn</B>.਀㰀䈀刀㸀吀栀攀渀Ⰰ 愀 昀椀渀愀氀 挀愀爀搀Ⰰ 昀愀挀攀 甀瀀㨀 琀栀攀 㰀䈀㸀刀椀瘀攀爀㰀⼀䈀㸀⸀ 
<BR>Between each event, players make their bets.਀㰀䈀刀㸀吀栀攀 漀戀樀攀挀琀椀瘀攀 椀猀 琀漀 愀挀栀椀攀瘀攀 琀栀攀 㰀䄀 䠀刀䔀䘀㴀∀樀愀瘀愀猀挀爀椀瀀琀㨀伀瀀攀渀圀椀渀㌀⠀✀瀀漀欀攀爀ⴀ栀愀渀搀猀⸀最椀昀✀⤀∀㸀㰀䈀㸀戀攀猀琀 㔀ⴀ挀愀爀搀 瀀漀欀攀爀 栀愀渀搀㰀⼀䈀㸀㰀⼀䄀㸀Ⰰ 甀猀椀渀最 礀漀甀爀 琀眀漀 挀愀爀搀猀 愀渀搀 琀栀攀 㔀 挀愀爀搀猀 瀀氀愀礀攀搀 昀愀挀攀 甀瀀⸀ 
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<P><HR><P>਀ऀ㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀∀㈀∀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀∀㔀∀ 䈀䜀䌀伀䰀伀刀㴀∀⌀䄀䄀䌀䌀䄀䄀∀㸀㰀吀䐀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀㸀 
	<B>the Flop</B></TD></TABLE>਀㰀吀䄀䈀䰀䔀㸀㰀吀䐀㸀匀甀瀀瀀漀猀攀 礀漀甀✀爀攀 栀漀氀搀椀渀最 琀栀攀 昀漀氀氀漀眀椀渀最 瀀愀椀爀 椀渀 礀漀甀爀 栀愀渀搀㨀 
 &nbsp; <B>5</B><FONT SIZE="+1" COLOR="red">♥</FONT>, <B>5</B><FONT SIZE="+1" COLOR="black">♣</FONT>਀㰀䈀刀㸀圀栀愀琀 愀爀攀 礀漀甀爀 挀栀愀渀挀攀猀 漀昀 最攀琀琀椀渀最 愀渀漀琀栀攀爀 㰀䈀㸀㔀㰀⼀䈀㸀Ⰰ 琀漀 洀愀欀攀 椀琀 ㌀ⴀ漀昀ⴀ愀ⴀ欀椀渀搀㼀 
<BR>The cards that can improve your hand are called <I>outs</I> ... and, in this case, your outs (to improve to 3-of-a-kind) are <B>5</B><FONT SIZE="+1" COLOR="red">♦</FONT> and <B>5</B><FONT SIZE="+1" COLOR="black">♠</FONT>.਀ 
<P>There are 52 cards in the deck and 2 are in your hand, leaving <FONT COLOR="blue"><B>50</B></FONT> cards unseen.਀㰀䈀刀㸀吀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 最攀琀琀椀渀最 漀渀攀 漀昀 礀漀甀爀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀最爀攀攀渀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 ∀漀甀琀猀∀ 椀猀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀最爀攀攀渀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 椀渀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㔀　㰀⼀䈀㸀㰀⼀䘀伀一吀㸀⸀ 
... and that's 4% <FONT COLOR="#555555" SIZE="-1">(since 100 * <FONT COLOR="green"><B>2</B></FONT> / <FONT COLOR="blue"><B>50</B></FONT> = 4%)</FONT>.਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀匀漀 椀琀✀猀 樀甀猀琀 ⠀渀甀洀戀攀爀 漀昀 漀甀琀猀⤀ ⼀ ⠀渀甀洀戀攀爀 漀昀 挀愀爀搀猀 甀渀猀攀攀渀⤀ 㼀㰀⼀䘀伀一吀㸀 
<BR>Yes, if there are <FONT COLOR="blue"><B>50</B></FONT> cards unseen.਀㰀䈀刀㸀䠀漀眀攀瘀攀爀Ⰰ 愀昀琀攀爀 㰀䈀㸀琀栀攀 䘀氀漀瀀㰀⼀䈀㸀 琀栀攀爀攀✀氀氀 戀攀 愀渀漀琀栀攀爀 ㌀ 挀愀爀搀猀 漀渀 琀栀攀 琀愀戀氀攀Ⰰ 猀漀 琀栀攀 渀甀洀戀攀爀 漀昀 挀愀爀搀猀 甀渀猀攀攀渀 椀猀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㐀㜀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀⸀ 
<BR>Suppose the flop doesn't have a <B>5</B><FONT SIZE="+1" COLOR="red">♦</FONT> or a <B>5</B><FONT SIZE="+1" COLOR="black">♠</FONT>.਀㰀䈀刀㸀吀栀攀渀 礀漀甀 渀攀攀搀 琀漀 挀愀氀挀甀氀愀琀攀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 最攀琀琀椀渀最 
one of these <FONT COLOR="green"><B>2</B></FONT> outs from the <FONT COLOR="blue"><B>47</B></FONT> unseen cards, and ...਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀䤀✀搀 猀愀礀 琀栀攀 挀栀愀渀挀攀猀 愀爀攀 ㈀ 漀甀琀 漀昀 㐀㜀⸀ 刀椀最栀琀㼀㰀⼀䘀伀一吀㸀 
<BR>Right, and that's about <FONT COLOR="green"><B></B></FONT> <FONT COLOR="magenta"><B>4.25</B></FONT>%.਀㰀倀㸀匀漀洀攀 氀椀欀攀 琀漀 渀漀琀攀 琀栀攀 ∀漀搀搀猀∀ 爀愀琀栀攀爀 琀栀愀渀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀⸀ 
<BR>If there are X ways to win and Y ways to lose, then the odds are Y-to-X or Y/X to 1.਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀吀栀愀琀✀猀 挀漀渀昀甀猀椀渀最℀㰀⼀䘀伀一吀㸀 
<BR>It's just saying for every Y ways to win there are X ways to lose ... and you normally say it as: <I>"The odds are 7-to-1"</I>.਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀䄀渀搀 琀栀愀琀✀搀 洀攀愀渀 琀栀愀琀 昀漀爀 攀瘀攀爀礀 㜀 眀愀礀猀 琀漀 眀椀渀 琀栀攀爀攀✀猀 ㄀ 眀愀礀 琀漀 氀漀猀攀⸀ 㰀⼀䘀伀一吀㸀 
<BR>Yes. Of <B>8</B> possible ways events can occur, <B>7</B> will be losses and <B>1</B> will be a win.਀㰀䈀刀㸀吀栀愀琀 洀攀愀渀猀 礀漀甀爀 眀椀渀渀椀渀最 瀀爀漀戀愀戀椀氀椀琀礀 椀猀 㰀䈀㸀㄀㰀⼀䈀㸀 ⼀ 㰀䈀㸀㠀㰀⼀䈀㸀 漀爀 ㄀㈀⸀㔀─⸀ 
਀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀 
<TABLE BORDER=1 CELLPADDING=5>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀夀漀甀爀 䠀愀渀搀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀琀栀攀 䘀氀漀瀀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀伀甀琀猀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀倀爀漀戀愀戀椀氀椀琀礀㰀⼀吀䐀㸀㰀吀刀㸀 
਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ 㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀㰀⼀吀䐀㸀 
<TD><CENTER>A<FONT SIZE="+1" COLOR="black">♠</FONT>, 3<FONT SIZE="+1" COLOR="black">♣</FONT>, J<FONT SIZE="+1" COLOR="red">♦</FONT></TD>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ 㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀㰀䈀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀琀漀 洀愀欀攀 ㌀ⴀ漀昀ⴀ愀ⴀ欀椀渀搀㰀⼀吀䐀㸀 
<TD><CENTER>2/47 = 4.25%</TD><TR>਀ 
<TD><CENTER>5<FONT SIZE="+1" COLOR="red">♥</FONT>, 5<FONT SIZE="+1" COLOR="black">♣</FONT></TD>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀䄀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀Ⰰ 㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ 䨀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀㰀⼀吀䐀㸀 
<TD><CENTER>5<FONT SIZE="+1" COLOR="black">♠</FONT><BR><FONT SIZE="-1">to make 3-of-a-kind</TD>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㄀⼀㐀㜀 㴀 ㈀⸀㄀㌀─㰀⼀吀䐀㸀㰀吀刀㸀 
਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ 㜀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀㰀⼀吀䐀㸀 
<TD><CENTER>6<FONT SIZE="+1" COLOR="black">♠</FONT>, 9<FONT SIZE="+1" COLOR="black">♣</FONT>, A<FONT SIZE="+1" COLOR="red">♦</FONT></TD>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㠀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀Ⰰ 㠀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ 㠀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ 㠀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀㰀䈀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀琀漀 洀愀欀攀 琀栀攀 椀渀猀椀搀攀 猀琀爀愀椀最栀琀㰀⼀吀䐀㸀 
<TD><CENTER>4/47 = 8.51%</TD><TR>਀ 
<TD><CENTER>J<FONT SIZE="+1" COLOR="black">♣</FONT>, K<FONT SIZE="+1" COLOR="red">♦</FONT></TD>਀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀䨀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ ㌀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀Ⰰ 㔀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀㰀⼀吀䐀㸀 
<TD><CENTER>J<FONT SIZE="+1" COLOR="red">♥</FONT>, J<FONT SIZE="+1" COLOR="black">♠</FONT>, K<FONT SIZE="+1" COLOR="black">♣</FONT>, K<FONT SIZE="+1" COLOR="red">♥</FONT>, K<FONT SIZE="+1" COLOR="black">♠</FONT>਀㰀䈀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀琀漀 椀洀瀀爀漀瘀攀 礀漀甀爀 栀愀渀搀㰀⼀吀䐀㸀 
<TD><CENTER>5/47 = 10.6%</TD><TR>਀ 
</TABLE>਀㰀䈀㸀吀愀戀氀攀 ㄀㨀㰀⼀䈀㸀 匀漀洀攀 瀀爀漀戀愀戀椀氀椀琀礀 挀愀氀挀甀氀愀琀椀漀渀猀㰀䈀刀㸀戀愀猀攀搀 甀瀀漀渀 渀攀攀搀椀渀最 㰀䈀㸀㰀唀㸀漀渀攀㰀⼀唀㸀㰀⼀䈀㸀 漀甀琀 琀漀 椀洀瀀爀漀瘀攀 礀漀甀爀 栀愀渀搀 
</TD></TABLE>਀㰀吀䄀䈀䰀䔀 圀䤀䐀吀䠀㴀㄀　　─㸀㰀吀䐀㸀 
<FONT COLOR="#000077">>So if odds are <B>Y / X</B>, then the probability of winning is <B>X / (X+Y)</B> .</FONT>਀㰀䈀刀㸀唀栀 ⸀⸀⸀ 礀攀猀Ⰰ 戀甀琀 椀昀 猀漀洀攀戀漀搀礀 猀愀礀猀 琀栀攀 漀搀搀猀 愀爀攀 㜀ⴀ琀漀ⴀ㄀ 椀琀✀猀 攀愀猀椀攀爀 樀甀猀琀 琀漀 挀愀氀挀甀氀愀琀攀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 愀猀 ㄀⼀⠀㜀⬀㄀⤀ 㴀 　⸀㄀㈀㔀⸀ 
<BR>Normally, you'd multiply by 100 changing it to 12.5%.਀㰀倀㸀䤀渀 漀甀爀 攀砀愀洀瀀氀攀Ⰰ 漀昀 琀栀攀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㐀㜀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 甀渀猀攀攀渀 挀愀爀搀猀Ⰰ 堀 㴀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀最爀攀攀渀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 椀猀 琀栀攀 渀甀洀戀攀爀 漀昀 挀愀爀搀猀 琀栀愀琀✀氀氀 最椀瘀攀 礀漀甀 礀漀甀爀 ㌀ⴀ漀昀ⴀ愀ⴀ欀椀渀搀⸀ 
<BR>The number of cards that'll "miss you" is then <FONT COLOR="blue"><B>47</B></FONT> - <FONT COLOR="green"><B>2</B></FONT> = <B>45</B>.਀吀栀攀渀 㰀䈀㸀夀㰀⼀䈀㸀 㴀 㰀䈀㸀㐀㔀㰀⼀䈀㸀⸀ 
<BR>The odds are then <B>Y</B>-to-<B>X</B> or <B>45</B> to <FONT COLOR="green"><B>2</B></FONT>   or <FONT COLOR="magenta"><B>22.5</B></FONT> to 1.਀ 
</TD><TD>਀ऀ㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀∀㈀∀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀∀㔀∀ 䈀䜀䌀伀䰀伀刀㴀∀⌀䐀䐀䘀䘀䐀䐀∀㸀㰀吀䐀㸀 
	If <B>P</B> is the probability of getting a winning card &nbsp; &nbsp; <FONT COLOR="#555555" SIZE="-1">(<U>Example</U>: P = 12.5%)</FONT>਀ऀ㰀䈀刀㸀愀渀搀 㰀䈀㸀娀㰀⼀䈀㸀 最椀瘀攀猀 琀栀攀 漀搀搀猀  ☀渀戀猀瀀㬀 ☀渀戀猀瀀㬀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀㰀唀㸀䔀砀愀洀瀀氀攀㰀⼀唀㸀㨀 娀 㴀 㜀 洀攀愀渀椀渀最 㜀ⴀ琀漀ⴀ㄀ 漀搀搀猀⤀㰀⼀䘀伀一吀㸀 
	<BR>then <B>P</B> = 100/(1+<B>Z</B>)  &nbsp; &nbsp; <FONT COLOR="#555555" SIZE="-1">(<U>Example</U>: P = 100/(1+7) = 12.5%)</FONT>਀ऀ㰀䈀刀㸀愀渀搀 㰀䈀㸀娀㰀⼀䈀㸀 㴀 ㄀　　⼀㰀䈀㸀倀㰀⼀䈀㸀 ⴀ ㄀   ☀渀戀猀瀀㬀 ☀渀戀猀瀀㬀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀㰀唀㸀䔀砀愀洀瀀氀攀㰀⼀唀㸀㨀 娀 㴀 ㄀　　⼀倀 ⴀ ㄀ 㴀 㜀⤀㰀⼀䘀伀一吀㸀 
	</TABLE>਀㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
<FONT COLOR="#000077">>Last time you got the probability as <FONT COLOR="magenta"><B>4.25</B></FONT>%.</FONT>਀㰀䈀刀㸀夀攀猀Ⰰ 愀渀搀 琀栀愀琀✀猀 ㄀⼀⠀㄀⬀漀搀搀猀⤀ 㴀 ㄀⼀⠀㄀⬀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀洀愀最攀渀琀愀∀㸀㰀䈀㸀㈀㈀⸀㔀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀⤀ 㴀 　⸀　㐀㈀㔀 漀爀  
<FONT COLOR="#555555" SIZE="-1">(multiplying by 100, to get a percentage)</FONT> <FONT COLOR="magenta"><B>4.25</B></FONT>%.਀ 
਀㰀倀㸀㰀䠀刀㸀㰀倀㸀 
	<TABLE BORDER="2" CELLPADDING="5" BGCOLOR="#AACCAA"><TD><FONT SIZE="+1">਀ऀ㰀䈀㸀琀栀攀 吀甀爀渀 愀渀搀 琀栀攀 刀椀瘀攀爀㰀⼀䈀㸀㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
<P>After <B>the Flop</B>, there are three cards face up on the table. That means there are now <FONT COLOR="blue"><B>47</B></FONT> cards unseen.਀㰀䈀刀㸀䄀昀琀攀爀 㰀䈀㸀琀栀攀 吀甀爀渀㰀⼀䈀㸀Ⰰ 琀栀攀爀攀 愀爀攀 昀漀甀爀 挀愀爀搀猀 昀愀挀攀 甀瀀 漀渀 琀栀攀 琀愀戀氀攀⸀ 吀栀愀琀 洀攀愀渀猀 琀栀攀爀攀 愀爀攀 渀漀眀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㐀㘀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 挀愀爀搀猀 甀渀猀攀攀渀⸀ 
<BR>After <B>the River</B>, there are five cards face up on the table. That means there are now <FONT COLOR="blue"><B>45</B></FONT> cards unseen.਀㰀䈀刀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀吀栀愀琀✀猀 㔀㈀Ⰰ 氀攀猀猀 琀栀攀 ㈀ 椀渀 礀漀甀爀 䠀愀渀搀Ⰰ 氀攀猀猀 琀栀攀 挀愀爀搀猀 昀愀挀攀 甀瀀Ⰰ 漀渀 琀栀攀 琀愀戀氀攀⸀⤀㰀⼀䘀伀一吀㸀 
਀㰀倀㸀匀甀瀀瀀漀猀攀 礀漀甀 栀漀氀搀 㰀䈀㸀㔀㰀⼀䈀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ 㰀䈀㸀㔀㰀⼀䈀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀 愀渀搀 㰀䈀㸀琀栀攀 䘀氀漀瀀㰀⼀䈀㸀 搀漀攀猀渀✀琀 栀愀瘀攀 攀椀琀栀攀爀  
the <B>5</B><FONT SIZE="+1" COLOR="red">♦</FONT> or the <B>5</B><FONT SIZE="+1" COLOR="black">♠</FONT>.਀ 
<BR>Suppose you want to calculate the probability of getting one of the <FONT COLOR="green"><B>2</B></FONT> cards that'll improve your hand on the next <U>two</U> cards laid face up.਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀䠀甀栀㼀㰀⼀䘀伀一吀㸀 
<BR>The probability of getting either the <B>5</B><FONT SIZE="+1" COLOR="red">♦</FONT> or the <B>5</B><FONT SIZE="+1" COLOR="black">♠</FONT>, immediately after the flop਀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀眀栀攀渀 琀栀攀爀攀 愀爀攀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㐀㜀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 挀愀爀搀猀 甀渀猀攀攀渀⤀㰀⼀䘀伀一吀㸀 
is <FONT COLOR="magenta"><B>4.25</B></FONT>% as we've noted before in Table 1. <FONT COLOR="#555555" SIZE="-1">(That's 2/47.)</FONT>਀㰀䈀刀㸀吀栀愀琀✀猀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 最攀琀琀椀渀最 漀渀攀 漀昀 礀漀甀爀 漀甀琀猀 漀渀 㰀䈀㸀琀栀攀 吀甀爀渀㰀⼀䈀㸀 㰀唀㸀愀渀搀㰀⼀唀㸀 㰀䈀㸀琀栀攀 刀椀瘀攀爀㰀⼀䈀㸀⸀ 
<BR>If there were just <U>one</U> card to go, you'd expect the probability to be half as much.਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀吀栀愀琀✀猀 㰀䤀㸀猀琀椀氀氀㰀⼀䤀㸀 挀漀渀昀甀猀椀渀最℀㰀⼀䘀伀一吀㸀 
<BR>Okay, we'll do it this way:਀ 
਀㰀倀㸀㰀䠀刀㸀㰀倀㸀 
	<TABLE BORDER="2" CELLPADDING="5" BGCOLOR="#AACCAA"><TD><FONT SIZE="+1">਀ऀ㰀䈀㸀䌀漀洀戀椀渀愀琀漀爀椀挀猀㰀⼀䈀㸀㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
<P>਀㰀唀䰀㸀 
<LI>If you have 52 cards and you ask: <I>"How many ways can I be dealt 2 cards?"</I>਀㰀䰀䤀㸀吀栀攀 愀渀猀眀攀爀 椀猀㨀  㔀㈀℀⼀⠀㈀℀ 㔀　℀⤀ 㴀 ⠀㔀㈀⤀⠀㔀㄀⤀⼀⠀㈀⤀⠀㄀⤀ 㴀 ㄀㌀㈀㘀Ⰰ 搀攀渀漀琀攀搀 戀礀 㰀匀唀䈀㸀㔀㈀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⸀⸀⸀ 爀攀愀搀 愀猀 㰀䤀㸀㔀㈀ⴀ䌀栀漀漀猀攀ⴀ㈀㰀⼀䤀㸀㰀⼀䘀伀一吀㸀⸀ 
<BR><FONT COLOR="#555555" SIZE="-1">Note that 52! = (52)(51)(50)...(3)(2)(1), the product of all integers from 1 to 52.</FONT>਀㰀䰀䤀㸀䤀渀 最攀渀攀爀愀氀Ⰰ 椀昀 礀漀甀 栀愀瘀攀 一 挀愀爀搀猀 愀渀搀 礀漀甀 愀猀欀㨀 㰀䤀㸀∀䠀漀眀 洀愀渀礀 眀愀礀猀 挀愀渀 䤀 戀攀 搀攀愀氀琀 䴀 挀愀爀搀猀∀㰀⼀䤀㸀Ⰰ 琀栀攀 愀渀猀眀攀爀 椀猀 㰀匀唀䈀㸀一㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀䴀㰀⼀匀唀䈀㸀⸀ 
</UL>਀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀匀漀 琀栀攀爀攀 愀爀攀 ㄀㌀㈀㘀 瀀漀猀猀椀戀氀攀 ㈀ⴀ挀愀爀搀 栀愀渀搀猀 搀攀愀氀琀 椀渀 吀攀砀愀猀 䠀漀氀搀✀攀洀㼀㰀⼀䘀伀一吀㸀 
<BR>Yes. In fact, there are 52 possibilities for the first card and <FONT COLOR="#555555" SIZE="-1">(having dealt that card)</FONT> 51 for the second so that makes (52)(51) = 2652.਀㰀䈀刀㸀䠀漀眀攀瘀攀爀Ⰰ 愀洀漀渀最 琀栀攀猀攀 ㈀㘀㔀㈀ 瀀漀猀猀椀戀椀氀椀琀椀攀猀 眀攀✀搀 昀椀渀搀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀爀攀搀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 椀搀攀渀琀椀挀愀氀 栀愀渀搀猀㨀 嬀㰀䈀㸀㔀㰀⼀䈀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ 㰀䈀㸀㔀㰀⼀䈀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀崀 
as well as [<B>5</B><FONT SIZE="+1" COLOR="black">♠</FONT>, <B>5</B><FONT SIZE="+1" COLOR="red">♦</FONT>].਀㰀䈀刀㸀匀椀渀挀攀 漀渀攀 搀漀攀猀渀✀琀 搀椀猀琀椀渀最甀椀猀栀 戀攀琀眀攀攀渀 琀栀攀猀攀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀爀攀搀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 栀愀渀搀猀Ⰰ 眀攀 搀椀瘀椀搀攀 ㈀㘀㔀㈀ 戀礀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀爀攀搀∀㸀㰀䈀㸀㈀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 最攀琀琀椀渀最 漀甀爀 ㄀㌀㈀㘀 ⸀⸀⸀ 愀渀搀 琀栀愀琀✀猀 㰀匀唀䈀㸀㔀㈀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀⸀ 
<P>Suppose we ask:  <I>"How many ways can I deal a pair?"</I>਀㰀唀䰀㸀 
<LI>There are 1326 ways to be dealt two cards, how many of these are pairs?਀㰀䰀䤀㸀吀栀攀爀攀 愀爀攀 昀漀甀爀 ㈀猀 椀渀 琀栀攀 搀攀挀欀Ⰰ ㈀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ ㈀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀Ⰰ ㈀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀Ⰰ ㈀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀⸀ 
<LI>We want to choose 2 of them. The number of ways is <SUB>4</SUB>C<SUB>2</SUB> = (4)(3)/2 = 6.਀㰀䰀䤀㸀䘀漀爀 愀氀氀 ㄀㌀ 挀愀爀搀猀Ⰰ 昀爀漀洀 琀栀攀 ㈀ 琀漀 琀栀攀 䄀Ⰰ 琀栀愀琀 洀攀愀渀猀 ㄀㌀⨀㘀 㴀 㜀㠀 瀀漀猀猀椀戀氀攀 瀀愀椀爀猀⸀ 
<LI>So, of the 1326 possible 2-card hands, 78 will be a pair.਀㰀䰀䤀㸀吀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 戀攀椀渀最 搀攀愀氀琀 愀 瀀愀椀爀 椀猀 琀栀攀渀㨀 㜀㠀⼀㄀㌀㈀㘀 㴀 　⸀　㔀㠀㠀 漀爀 㔀⸀㠀㠀─⸀ 
<BR><FONT COLOR="#555555" SIZE="-1">The odds are then: 100/probability - 1 = 100/5.88 - 1 = 16, hence 16-to-1.</FONT>਀㰀⼀唀䰀㸀 
<FONT COLOR="#000077">>Isn't there some magic (and simple!) formula?</FONT>਀㰀唀䰀㸀 
<LI>You calculate the total number of ways that something can happen <FONT COLOR="#555555" SIZE="-1">(like 1326 ways to deal 2 cards)</FONT>.਀㰀䰀䤀㸀䘀爀漀洀 琀栀攀猀攀Ⰰ 礀漀甀 椀搀攀渀琀椀昀礀 琀栀攀 渀甀洀戀攀爀 漀昀 眀愀礀猀 琀栀愀琀 猀漀洀攀琀栀椀渀最 㰀唀㸀猀瀀攀挀椀愀氀㰀⼀唀㸀 挀愀渀 栀愀瀀瀀攀渀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀氀椀欀攀 㜀㠀 眀愀礀猀 琀漀 最攀琀 愀 瀀愀椀爀⤀㰀⼀䘀伀一吀㸀⸀ 
<LI>You divide the latter by the former <FONT COLOR="#555555" SIZE="-1">(like 78/1326 to get 5.88% probability of something <U>special</U> happening)</FONT>.਀㰀⼀唀䰀㸀 
Okay, let's do something more complicated:਀㰀唀䰀㸀 
<LI>Suppose there are 4 outs and you want 1 of these 4 outs to improve your hand.਀㰀䰀䤀㸀䄀昀琀攀爀 琀栀攀 㰀䈀㸀䘀氀漀瀀㰀⼀䈀㸀Ⰰ 琀栀攀爀攀 愀爀攀 㐀㜀 挀愀爀搀 甀渀猀攀攀渀⸀ 
<LI>The number of possible combinations of 2 cards for the <B>Turn</B> and the <B>River</B> is <SUB>47</SUB>C<SUB>2</SUB>.਀㰀䰀䤀㸀䠀漀眀 洀愀渀礀 漀昀 琀栀攀猀攀 眀椀氀氀 栀愀瘀攀 ㄀ 漀昀 琀栀攀 㐀 漀甀琀猀㼀 
<LI>To calculate this, we remove the 4 outs, leaving 47 - 4 = 43 cards.਀㰀䰀䤀㸀吀栀攀爀攀 愀爀攀 㰀匀唀䈀㸀㐀㌀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 眀愀礀猀 琀漀 搀攀愀氀 ㈀ 挀愀爀搀猀 昀爀漀洀 琀栀攀猀攀 㐀㌀ 挀愀爀搀猀 ⸀⸀⸀ 愀渀搀 一伀一䔀 眀椀氀氀 栀愀瘀攀 愀渀礀 漀昀 琀栀攀 㐀 漀甀琀猀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀戀攀挀愀甀猀攀 眀攀✀瘀攀 爀攀洀漀瘀攀搀 琀栀攀洀⤀㰀⼀䘀伀一吀㸀⸀ 
<LI>So how many of the 2 cards (from the <B>Turn</B> and the <B>River</B>) WILL have 1 of the 4 outs?਀㰀䰀䤀㸀䤀琀✀猀 㰀匀唀䈀㸀㐀㜀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 ⴀ 㰀匀唀䈀㸀㐀㌀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 㴀 ⠀琀漀琀愀氀 ㈀ⴀ挀愀爀搀 搀攀愀氀猀⤀ ⴀ ⠀琀栀漀猀攀 琀栀愀琀 搀漀 一伀吀 栀愀瘀攀 愀渀 漀甀琀 挀愀爀搀⤀⸀ 
<LI>The probability is then: [ <SUB>47</SUB>C<SUB>2</SUB> - <SUB>43</SUB>C<SUB>2</SUB>] / <SUB>47</SUB>C<SUB>2</SUB> = (number with at least 1 of the outs) / (total 2-card deals).਀㰀䰀䤀㸀吀栀愀琀✀猀 愀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 ㄀ ⴀ 㰀匀唀䈀㸀㐀㌀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 ⼀ 㰀匀唀䈀㸀㐀㜀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 㴀 ㄀ ⴀ 　⸀㠀㌀㔀 㴀 　⸀㄀㘀㔀 漀爀 ㄀㘀⸀㔀─⸀  
</UL>਀㰀吀䄀䈀䰀䔀㸀㰀吀䐀㸀 
<FONT COLOR="#000077">>I think there's a magic formula there.</FONT>਀㰀䈀刀㸀夀攀猀Ⰰ 琀栀攀爀攀 愀爀攀 愀 挀漀甀瀀氀攀 愀渀搀 琀栀攀礀 最漀 氀椀欀攀 琀栀椀猀㨀 
	<TABLE BORDER="2" CELLPADDING="5" BGCOLOR="#DDFFDD" ><TD>਀ऀ㰀䈀㸀䄀昀琀攀爀 琀栀攀 䘀氀漀瀀㨀㰀⼀䈀㸀 
	<BR>There are 47 cards unseen and there are <B>n</B> outs which will improve your hand <BR><FONT COLOR="#555555" SIZE="-1"><U>Example</U>: n = 3 outs</FONT>.਀ऀ㰀䈀刀㸀䘀爀漀洀 琀栀攀 渀攀砀琀 ㈀ 挀愀爀搀猀 搀攀愀氀琀 ⠀㰀䈀㸀吀甀爀渀㰀⼀䈀㸀 ⬀ 㰀䈀㸀刀椀瘀攀爀㰀⼀䈀㸀⤀Ⰰ 礀漀甀 眀愀渀琀 ㄀ 漀甀琀 漀昀 琀栀攀 渀 漀甀琀猀 㰀䈀刀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㰀唀㸀䔀砀愀洀瀀氀攀㰀⼀唀㸀㨀 ㄀ 漀甀琀 漀昀 渀 㴀 ㌀ 漀甀琀猀㰀⼀䘀伀一吀㸀⸀ 
	 <BR>The probability of getting 1 of the n outs in the next 2 cards is: ਀ऀ㰀䈀刀㸀㰀䈀㸀倀爀漀戀愀戀椀氀椀琀礀㰀⼀䈀㸀 㴀 ㄀ ⴀ 㰀匀唀䈀㸀⠀㐀㜀ⴀ渀⤀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 ⼀ 㰀匀唀䈀㸀㐀㜀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 
	  <BR><FONT COLOR="#555555" SIZE="-1"><U>Example</U>: 1 - <SUB>44</SUB>C<SUB>2</SUB> / <SUB>47</SUB>C<SUB>2</SUB> = 0.125 or 12.5%</FONT>.਀ऀ㰀⼀吀䄀䈀䰀䔀㸀 
਀ऀ㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀∀㈀∀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀∀㔀∀ 䈀䜀䌀伀䰀伀刀㴀∀⌀䐀䐀䘀䘀䐀䐀∀㸀㰀吀䐀㸀 
	<B>After the Flop + Turn:</B>਀ऀ㰀䈀刀㸀吀栀攀爀攀 愀爀攀 㐀㘀 挀愀爀搀猀 甀渀猀攀攀渀 愀渀搀 琀栀攀爀攀 愀爀攀 㰀䈀㸀渀㰀⼀䈀㸀 漀甀琀猀 眀栀椀挀栀 眀椀氀氀 椀洀瀀爀漀瘀攀 礀漀甀爀 栀愀渀搀 㰀䈀刀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㰀唀㸀䔀砀愀洀瀀氀攀㰀⼀唀㸀㨀 渀 㴀 ㌀ 漀甀琀猀㰀⼀䘀伀一吀㸀⸀ 
	<BR>From the next card dealt (the <B>River</B>), you want 1 out of the n outs <BR><FONT COLOR="#555555" SIZE="-1"><U>Example</U>: 1 out of n = 3 outs</FONT>.਀ऀ 㰀䈀刀㸀吀栀攀 瀀爀漀戀愀戀椀氀椀琀礀 漀昀 最攀琀琀椀渀最 ㄀ 漀昀 琀栀攀 渀 漀甀琀猀 椀渀 琀栀攀 渀攀砀琀 挀愀爀搀 椀猀㨀  
	<BR><B>Probability</B> = 1 - <SUB>(46-n)</SUB>C<SUB>1</SUB> / <SUB>46</SUB>C<SUB>1</SUB>਀ऀ  㰀䈀刀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㰀唀㸀䔀砀愀洀瀀氀攀㰀⼀唀㸀㨀 ㄀ ⴀ 㰀匀唀䈀㸀㐀㌀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㄀㰀⼀匀唀䈀㸀 ⼀ 㰀匀唀䈀㸀㐀㘀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀 㴀 　⸀　㘀㔀 漀爀 㘀⸀㔀─㰀⼀䘀伀一吀㸀⸀ 
	</TABLE>਀ 
</TD><TD><CENTER>਀㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀㄀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀伀甀琀猀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀ 挀愀爀搀猀 琀漀 最漀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀ 挀愀爀搀 琀漀 最漀㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀⸀㌀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀⸀㈀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㠀⸀㐀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀⸀㌀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㈀⸀㔀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㘀⸀㔀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㘀⸀㔀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㠀⸀㜀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㔀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀　⸀㐀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀　⸀㤀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㘀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㐀⸀㄀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㌀⸀　─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㜀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㜀⸀㠀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㔀⸀㈀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㠀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㄀⸀㔀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㜀⸀㐀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㤀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㔀⸀　─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㤀⸀㘀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀　㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㠀⸀㐀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㄀⸀㜀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㄀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㄀⸀㜀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㌀⸀㤀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㈀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㔀⸀　─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㘀⸀㄀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㌀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㠀⸀㄀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀㠀⸀㌀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㐀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㔀㄀⸀㈀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀　⸀㐀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㔀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㔀㐀⸀㄀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㈀⸀㘀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㘀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㔀㜀⸀　─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㐀⸀㠀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㜀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㔀㤀⸀㠀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㜀⸀　─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㠀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㘀㈀⸀㐀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㌀㤀⸀㄀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㄀㤀㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㘀㔀⸀　─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㄀⸀㌀─㰀⼀吀䐀㸀㰀吀刀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㈀　㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㘀㜀⸀㔀─㰀⼀吀䐀㸀㰀吀䐀㸀㰀䌀䔀一吀䔀刀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㐀㌀⸀㔀─㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
<B>Table 2:</B></TD></TABLE>਀ 
<TABLE><TD>਀一漀眀 氀漀漀欀 挀愀爀攀昀甀氀氀礀 愀琀 吀愀戀氀攀 ㈀⸀ 一漀琀椀挀攀 琀栀愀琀 琀栀攀 瀀爀漀戀愀戀椀氀椀琀椀攀猀Ⰰ 眀栀攀渀 琀栀攀爀攀 愀爀攀 ㈀ 挀愀爀搀猀 琀漀 最漀Ⰰ 椀猀 爀漀甀最栀氀礀 㐀砀 琀栀攀 漀甀琀猀⸀ 
<BR>That's because of the following (when there are n outs):਀㰀唀䰀㸀 
<LI>Probability = 1 - <SUB>(47-n)</SUB>C<SUB>2</SUB> / <SUB>47</SUB>C<SUB>2</SUB> = 1 - [(47-n)(46-n)]/[(47)(46)] = 1 - (1-n/47)(1-n/46) = n (1/47 + 1/46) - n<SUP>2</SUP>/(47)(46)਀㰀䰀䤀㸀吀漀 最攀琀 愀 瀀攀爀挀攀渀琀愀最攀Ⰰ 洀甀氀琀椀瀀氀礀 戀礀 ㄀　　 愀渀搀 最攀琀㨀 
<BR> Probability = n (100/47 + 100/46) + 0.046 n<SUP>2</SUP> &#8776; 4.3 n &nbsp; &nbsp; <FONT COLOR="#555555" SIZE="-1">... when n (hence n<SUP>2</SUP>) isn't too large.</FONT>਀㰀⼀唀䰀㸀 
</TD><TD><IMG SRC="poker-1.gif"></TD></TABLE>਀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀䔀愀爀氀椀攀爀Ⰰ 礀漀甀 最漀琀 ㈀⼀㐀㜀 愀渀搀 渀漀眀 礀漀甀 最攀琀 ⸀⸀⸀㰀⼀䘀伀一吀㸀 
<BR>And now I get 1 - <SUB>46</SUB>C<SUB>2</SUB> / <SUB>47</SUB>C<SUB>2</SUB> = 1 - 45/47 = 2/47.਀㰀䈀刀㸀一攀愀琀Ⰰ 攀栀㼀  ☀渀戀猀瀀㬀 㰀䤀䴀䜀 匀刀䌀㴀∀猀洀椀氀攀礀㄀㐀⸀最椀昀∀㸀 
<P><FONT COLOR="#000077">>When you say the dealer has 47 cards, you're assumuing I'm the only player, right?</FONT>਀㰀䈀刀㸀䜀漀漀搀 瀀漀椀渀琀⸀  
਀ 
<P><HR><P>਀ऀ㰀吀䄀䈀䰀䔀 䈀伀刀䐀䔀刀㴀∀㈀∀ 䌀䔀䰀䰀倀䄀䐀䐀䤀一䜀㴀∀㔀∀ 䈀䜀䌀伀䰀伀刀㴀∀⌀䄀䄀䌀䌀䄀䄀∀㸀㰀吀䐀㸀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀㸀 
	<B>Other players</B></TD></TABLE>਀㰀倀㸀䤀昀 琀栀攀 搀攀愀氀攀爀 搀攀愀氀猀 ㈀ 挀愀爀搀猀 琀漀 礀漀甀Ⰰ 栀攀✀氀氀 栀愀瘀攀 㔀　 氀攀昀琀⸀ 
<BR>However, before <B>the Flop</B>, he'll deal 2 cards to each of the your opponents. ਀㰀䈀刀㸀匀漀Ⰰ 椀昀 琀栀攀爀攀 愀爀攀 洀 漀瀀瀀漀渀攀渀琀猀Ⰰ 栀漀眀 洀愀渀礀 眀愀礀猀 挀愀渀 ㈀ 挀愀爀搀猀 戀攀 搀攀愀氀琀 琀漀 攀愀挀栀㼀 
<BR>There are <SUB>50</SUB>C<SUB>2</SUB> ways to deal the first 2 and, for <I>each</I> of these, there are <SUB>48</SUB>C<SUB>2</SUB> to deal the next 2, then <SUB>46</SUB>C<SUB>2</SUB> <I>etc. etc.</I>਀㰀䈀刀㸀匀漀Ⰰ 昀漀爀 洀 㴀 ㌀ 漀瀀瀀漀渀攀渀琀猀Ⰰ 琀栀愀琀✀氀氀 戀攀 ⠀㰀匀唀䈀㸀㔀　㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀⤀⠀㰀匀唀䈀㸀㐀㠀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀⤀⠀㰀匀唀䈀㸀㐀㘀㰀⼀匀唀䈀㸀䌀㰀匀唀䈀㸀㈀㰀⼀匀唀䈀㸀⤀ 㴀 ㄀Ⰰ㐀㌀　Ⰰ㄀㘀㌀Ⰰ　　　⸀ 
<P><FONT COLOR="#000077">>Mamma mia!</FONT>਀㰀䈀刀㸀䠀漀氀搀 漀渀℀ 
<BR>Suppose the hands dealt to the 3 other players are:਀㰀䈀刀㸀嬀䨀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀攀㰦⼀䘀伀一吀㸀Ⰰ 㤀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀崀  嬀㘀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀挀㰦⼀䘀伀一吀㸀Ⰰ 㤀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀崀  
 [2<FONT SIZE="+1" COLOR="red">♥</FONT>, A<FONT SIZE="+1" COLOR="red">♦</FONT>].਀㰀䈀刀㸀圀栀攀渀 眀攀 挀漀甀渀琀攀搀 甀瀀 琀漀 ㄀Ⰰ㐀㌀　Ⰰ㄀㘀㌀Ⰰ　　　Ⰰ 琀栀愀琀 渀甀洀戀攀爀 椀渀挀氀甀搀攀搀 愀氀氀 漀昀 琀栀攀 爀攀愀爀爀愀渀最攀洀攀渀琀猀 漀昀 琀栀攀猀攀 ㌀ 栀愀渀搀猀⸀ 
<P><FONT COLOR="#000077">>Huh?</FONT>਀㰀䈀刀㸀䤀琀 挀漀甀渀琀攀搀 
[J<FONT SIZE="+1" COLOR="red">♥</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>] [6<FONT SIZE="+1" COLOR="black">♣</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>] [2<FONT SIZE="+1" COLOR="red">♥</FONT>, A<FONT SIZE="+1" COLOR="red">♦</FONT>]਀愀渀搀  
[6<FONT SIZE="+1" COLOR="black">♣</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>] [J<FONT SIZE="+1" COLOR="red">♥</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>] [2<FONT SIZE="+1" COLOR="red">♥</FONT>, A<FONT SIZE="+1" COLOR="red">♦</FONT>]਀愀渀搀 
 [2<FONT SIZE="+1" COLOR="red">♥</FONT>, A<FONT SIZE="+1" COLOR="red">♦</FONT>] [6<FONT SIZE="+1" COLOR="black">♣</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>] [J<FONT SIZE="+1" COLOR="red">♥</FONT>, 9<FONT SIZE="+1" COLOR="black">♠</FONT>]਀㰀䤀㸀攀琀挀⸀ 攀琀挀⸀㰀⼀䤀㸀 愀猀 搀椀猀琀椀渀挀琀⸀  
<BR>Clearly, it makes no difference which opponent holds which hand ... so they shouldn't be counted as distinct possibilities.਀㰀䈀刀㸀匀椀渀挀攀 琀栀攀爀攀 愀爀攀 ㌀⨀㈀⨀㄀ 㴀 ㌀℀ 㴀 㘀 瀀漀猀猀椀戀氀攀 爀攀愀爀爀愀渀最攀洀攀渀琀猀 昀漀爀 攀愀挀栀 猀攀琀 漀昀 ㌀Ⰰ 琀栀攀 渀甀洀戀攀爀 ㄀Ⰰ㐀㌀　Ⰰ㄀㘀㌀Ⰰ　　　 椀猀 㘀砀 琀漀漀 氀愀爀最攀 ⸀⸀⸀ 猀漀 眀攀 搀椀瘀椀搀攀 戀礀 㘀⸀ 
<BR>In fact, for m opponents, we'd divide by m! = 1*2*3*...*m. &nbsp;  <FONT COLOR="#555555" SIZE="-1">(In our m=3 example, that's 1*2*3 = 6.)</FONT>਀㰀倀㸀伀欀愀礀Ⰰ 猀漀 眀攀 栀愀瘀攀㨀 
	<TABLE BORDER="2" CELLPADDING="5" BGCOLOR="#DDFFDD"><TD>਀ऀ䘀爀漀洀 㔀　 挀愀爀搀猀Ⰰ 琀栀攀 渀甀洀戀攀爀 漀昀 搀椀昀昀攀爀攀渀琀 眀愀礀猀 琀漀 搀攀愀氀 㰀䈀㸀洀㰀⼀䈀㸀 猀攀琀猀 漀昀 ㈀ 挀愀爀搀猀 㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀⠀琀漀 洀 漀瀀瀀漀渀攀渀琀猀⤀㰀⼀䘀伀一吀㸀 椀猀㨀 
	<BR><B> (<SUB>50</SUB>C<SUB>2</SUB>) (<SUB>48</SUB>C<SUB>2</SUB>) (<SUB>46</SUB>C<SUB>2</SUB>) ... (<SUB>52-2m</SUB>C<SUB>2</SUB>) / m! </B>਀ऀ㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
<FONT COLOR="#000077">>Okay, suppose there are 9 opponents and I'd like to know all the possibilities so that ...</FONT>਀㰀䈀刀㸀䄀渀猀眀攀爀㨀 伀瘀攀爀 ㈀　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　⸀ 
<P><FONT COLOR="#000077">>Yeah, but if I had a computer I'd just ...</FONT>਀㰀䈀刀㸀䤀昀 礀漀甀爀 挀漀洀瀀甀琀攀爀 挀漀甀氀搀 挀愀氀挀甀氀愀琀攀 琀栀攀 瀀漀猀猀椀戀椀氀椀琀椀攀猀 愀琀 琀栀攀 爀愀琀攀 漀昀 ㄀　Ⰰ　　　Ⰰ　　　Ⰰ　　　Ⰰ　　　 瀀攀爀 猀攀挀漀渀搀Ⰰ 椀琀✀猀 琀愀欀攀 漀瘀攀爀 㘀　　 洀椀氀氀椀漀渀 礀攀愀爀猀⸀ 
<BR>In real cases, one wouldn't bother calculating all the possible combinations that would beat <U>your</U> hand ... else you'll be late for supper.਀㰀䈀刀㸀䤀渀猀琀攀愀搀Ⰰ 漀渀攀 椀洀愀最椀渀攀猀 琀栀漀甀猀愀渀搀猀 ⠀洀椀氀氀椀漀渀猀㼀⤀ 漀昀 栀愀渀搀猀 戀攀椀渀最 搀攀愀氀琀 琀漀 礀漀甀爀 漀瀀瀀漀渀攀渀琀猀 愀渀搀 猀攀攀椀渀最 眀栀愀琀 瀀攀爀挀攀渀琀愀最攀 戀攀愀琀 礀漀甀爀 栀愀渀搀⸀ 
<P><FONT COLOR="#000077">>And that's the probability of being beaten by an opponent?</FONT>਀㰀䈀刀㸀䄀瀀瀀爀漀砀椀洀愀琀攀氀礀⸀ 䤀琀✀猀 猀漀爀琀愀 氀椀欀攀 瀀漀氀氀椀渀最 ㄀　　　 瀀攀漀瀀氀攀 琀漀 猀攀攀 眀栀椀挀栀 瀀爀攀昀攀爀 挀栀漀挀漀氀愀琀攀 椀挀攀 挀爀攀愀洀 琀漀 瘀愀渀椀氀氀愀⸀ 
<BR>Then you use that percentage to gauge the entire population of millions.਀㰀䈀刀㸀伀爀 椀琀✀猀 氀椀欀攀 搀漀椀渀最 愀 昀攀眀 琀栀漀甀猀愀渀搀 䴀漀渀琀攀 䌀愀爀氀漀 猀椀洀甀氀愀琀椀漀渀猀 琀漀 猀攀攀 栀漀眀 氀漀渀最 礀漀甀爀 瀀漀爀琀昀漀氀椀漀 洀椀最栀琀 氀愀猀琀 椀昀 礀漀甀 眀椀琀栀搀爀愀眀 砀─ 攀愀挀栀 礀攀愀爀⸀ 
<BR>Or it's like ...਀㰀倀㸀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀　　　　㜀㜀∀㸀㸀㰀䈀㸀伀欀愀礀℀㰀⼀䈀㸀 䤀 最攀琀 椀琀⸀㰀⼀䘀伀一吀㸀 
<BR>Good. ਀ 
਀ 
<P><P><IMG SRC="in_progress.gif">਀㰀⼀吀䐀㸀㰀⼀吀䄀䈀䰀䔀㸀 
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਀㰀℀ⴀⴀ 
<FONT SIZE="+1" COLOR="red">♥</FONT>਀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀爀攀搀∀㸀昀㰦⼀䘀伀一吀㸀 
<FONT SIZE="+1" COLOR="black">♣</FONT>਀㰀䘀伀一吀 匀䤀娀䔀㴀∀⬀㄀∀ 䌀伀䰀伀刀㴀∀戀氀愀挀欀∀㸀怀㰦⼀䘀伀一吀㸀 
<SUB>N</SUB>C<SUB>M</SUB>਀ⴀⴀ㸀 
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<FONT COLOR="#000077"></FONT>਀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀⌀㔀㔀㔀㔀㔀㔀∀ 匀䤀娀䔀㴀∀ⴀ㄀∀㸀㰀⼀䘀伀一吀㸀 
<FONT COLOR="green"><B></B></FONT>਀㰀䘀伀一吀 䌀伀䰀伀刀㴀∀戀氀甀攀∀㸀㰀䈀㸀㰀⼀䈀㸀㰀⼀䘀伀一吀㸀 
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