Had this situation last weekend. Last hand of an early round in a $750k tournament. Two people advance. Rules are no surrender, min bet 500, no max bet. We started with 10k (not relevant, but interesting considering stack sizes). Land hand betting order: BR1 (me) 30,000 BR3 25,000 BR4 24,000 BR2 27,000 (numbers are approximate) Obviously BR2 is in the best shape here. I took the 2nd low -- 4900 bet -- on the theory that if I bet big, even 15,000, I'm going to completely lose the low (at least two people will hold back more chips than me). Part of it was the likely order. If I bet more than 4900, BR3 realizes he can take the 2nd low and takes it. BR4 probably bets big. BR2 then takes 1st low or 2nd low. And I have to win hand . Did I make the right bet? I think I did. Of course, BR3 got a BJ and BR2 won the hand with a 20 -- I had a 16 that I busted, so I was out.
It was irrelevant. It was a free hit, but really it was a free stay too. My hand became irrelevant when BR3 went all in and got a BJ, and BR2 basically went all in and got a hard 20. BR4 had also gone all in but had a 19, I think, and there was no chance on earth he was going to hit it (and he had me covered either way if he won). I couldn't double to pass BR2, and losing didn't cost me anything vis-a-vis anybody. I still had the low, which required dealer to make 21.
Take the first low. With 4 players remaining and with 2 advancing a BR1 betting first should take the first low with a 58.88% chance to advance. By taking the first high he has a 49.96% chance to advance if the other players take the lows but bet enough to pass a BR1 push. ..................................BlueLight
Wong Book Wrong This disagrees with Wong's CTS, which states in Chapter 6, Table 3 that when two advance from four first high is best. First low is second-best. What was your methodology in obtaining your numbers? Are any of the other entries in Table 3 incorrect? Not that I'm disagreeing with you. I'm just Curious George as to how you obtained this data. Here is a summary of Table 3, which lists rank order goals in two-advance games for a given number of players when considering your bet: 3 Players Best - 2nd high/2nd low 2nd - 1st low 3rd - 1st high 4 Players Best - 1st high 2nd - 1st low 3rd - 2nd high 5 or More Players Best - 1st high 2nd - 2nd high 3rd - 1st low Note that the section warns against seeking second low unless there are only three players AND you can also get second high.
Use the 3 Player and 4 Player Outcome Tables In November 2008 and February 2009 I posted 3 and 4 player W P L outcome probabilities for 3 and 4 players. The tables agree with Wong's figures on p.131 of his book. Below I give figures for 4 players when dealer hits soft 17 (the tables I posted in 2009 were for dealer standing on soft 17). Wong gives figures to only 2 decimal places and sometimes he has to combine several probabilities together to get a useable result. Examples: For 4 players the probability of all 4 players winning is given as .19 while I give .195225. For 4 players with 3 winning and the other losing or pushing is given as .13 I break it apart as ; W W W P............ 0.004204...........4 ways...... 0.016816 W W W L............ 0.028202...........4 ways...... 0.112808 ........................................................Sum = 0.129524 Agrees with Wong's .13 In the table below the probability of 2 players winning and 2 pushing (WWPP) has 6 ways as shown: ...............W W P P ...............W P W P ...............W P P W ...............P W W P ...............P W P W ...............P P W W Wong only gives 2 players winning and 2 players losing or pushing; that is combining: WWPP WWPL WWLL Below is a way to compute the BR1 probability of winning when BR1 takes the FIRST HIGH. WWWW.... .195225...1... .195225..............1... .195225 WWWP..... .004204...4... .016816..............3... .012612 WWWL..... .028292...4... .112808..............3... .084606 WWPP...... .001212...6... .007272..............3... .003636 WWPL...... .003858..12... .046296..............6... .023148 WWLL....... .019851...6... .119106..............3... .059553 WPPP....... .000444...4... .001776...............4... .001776 WPPL....... .001542..12... .018504...............9... .013878 WPLL....... .006030..12... .072360...............6... .036180 WLLL....... .030070....4... .120280...............1... .030070 PPPP........ .000277....1... .000277...............1... .000277 PPPL........ .001058....4... .004232...............3... .003174 PPLL........ .004421....6... .026526...............3... .013263 PLLL........ .022206....4... .088824...............1... .022205 LLLL........ .169698....1... .169698...............0... .000000 .........................Sum = 1.000000.........Sum = 0.499604 The first column of sum gives the total probabilities. The second column gives only the sum of probabilities where BR1 advances when he takes the FIRST HIGH. Thus by taking the FIRST HIGH and no LOW chance, BR1 will advance 49.96% of the time. When BR1 takes the FIRST LOW the three player outcome table can be used since it doesn't matter whether BR1 wins or loses his small bet. *WWW...... .22763.....1... .22763................0... .00000 *WWP....... .00927.....3... .02781................0... .00000 *WWL....... .05191.....3... .15573................0... .00000 *WPP........ .00320.....3... .00960................3... .00960 *WPL........ .01143.....6... .06858................6... .06858 *WLL........ .05595.....3... .16785................3... .16785 *PPP......... .00178.....1... .00178................1... .00178 *PPL......... .00702.....3... .02106................3... .02106 *PLL......... .03266.....3... .09798................3... .09798 *LLL......... .22198.....1... .22198................1... .22198 ........................Sum = 1.00000...........Sum = 0.58883 By BR1 taking the first LOW and no chance for a HIGH if 2 other players win then BR1 will advance 58.88% of the time. I disagree with Wong's table 3 about taking the high as best, unless BR1 bets last and can get a high and a low over other players. ....................................BlueLight
