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being divided 3–0. The first card is already in, let us say, the East hand. Now he has only 12 vacant places so the probability of that hand getting the second of the three cards is 12/(12 + 13). This must be multiplied by the initial 1/2 probability to find the probability of East having both of the first two cards. Now let's deal out the third (and last) of the missing cards. By this time, East has only 11 vacant places, while West still has 13. The probability of East getting all three of the missing cards is 1/2 × 12/25 × 11/24 which is exactly 0.11, which is the value that we see in the fourth row of the table (3 - 0 : 0.22 : 2 : 0.11).
104:, each of four hands comprises thirteen cards and one may say there are thirteen vacant places in each hand. The probability that a particular card lies in a particular hand is one-quarter, or 13/52, the proportion of vacant places in that hand. From the perspective of a player who sees one hand, the probable lie of a missing card in a particular one of the other hands is one-third. In Contract bridge, once the play commences, the dummy is exposed and so, for any player, there are only two unseen hands where a card may lie.
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115:. Essentially, as the lie of some cards becomes known – especially as the entire distributions of some suits become known – the odds on location of any other particular card remain proportional to the dwindling numbers of unidentified cards in all hands, i.e. to the numbers of so-called vacant places.
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2. The last of the three spot cards appears on our right leaving one outstanding heart, the queen. Because no one would play the queen while holding a spot card too, we have learned nothing about the location of the queen directly, only the distribution of the three spot cards, one at left and two at
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Q may reside at left and eleven vacant places at right. If the queen lies in 12 of the 23 vacant places, at left, we win by playing the ace; the queen drops. In 11 of the 23 vacant places, we win by playing the jack and then the ace, dropping the queen at right on the next heart trick. Thus the odds
386:
we have five spades between dummy and hand leaving eight for the opponents. We may infer that LHO has six spades and RHO two. (This is not certain; occasionally the spades lie seven and one or five and three. If six and two, that leaves seven and eleven vacant places for the other three suits.) The
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probabilities). When we "deal" out the first card of the three, we can put it in either hand. Each hand, by definition, has 13 vacant places, so it is a toss-up which hand it goes into (13/26 = 50% for either hand). Now let us suppose that we want to know the probability of the suit
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Q at left and nine vacant places at right. The odds are now 6 to 9 against the queen at left, and against winning if we play the ace. The proportion 9/15 = 60% is the probability that RHO holds the queen and playing the jack will win the suit.
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in favor of playing the ace are 12 to 11; the ace is a slight favorite to win an extra trick, i.e. to win five tricks in hearts. The proportion 12/23 = 52.174% is exactly the probability that appears in standard catalogs of suit combinations.
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First, suppose we know nothing about the other suits, probably because the opponents did not bid. Then we know only the one small heart observed at left and the two observed at right. That leaves twelve "vacant places" where
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is a simple method for estimating the probable location of any particular card in the two unseen hands. It can be used both to aid in a decision at the table and to derive the entire suit division probability table.
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This amount has to be multiplied by 6, exactly the ways the 2–2 distribution can show up, the combination of getting 2 cards over 4. The final probability of a 2–2 split is then 0.067826 * 6 = 0.4069565217
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Now, let's calculate the individual probability of a 2–2 split when missing four cards (the following row in the table). This time, proceeding similarly to before, the calculation is:
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We are the declarer in a heart contract with trump suit combination Kxxx in dummy and AJxxx in hand (see figure). There are four heart cards missing, the queen and three spot cards or
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hearts, that is to say every heart but the one we are still seeking. If we were missing a total of five heart cards, then a vacant places calculation could not be applied.
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heart suit combination and play are as discussed above: the combination in the figure; we lead small to the king while both opponents follow low, and lead
478:. Let's suppose that we are missing three cards in the suit and we know nothing about the distribution of other suits (i.e. we are looking for the
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126:. For a good background to bridge probabilities, and vacant places in particular, see Kelsey; see also the Official Encyclopedia of Bridge.
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Let us imagine that we are required to build up a set of probability tables to help show how a suit might be splitting, for example, the
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The principle of vacant places is a rule for updating those uniform probabilities as one learns about the deal during the
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Note, however, that this calculation is only available in the heart suit because we have accounted for all the
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Qxxx. We play small to the king as both opponents follow low and lead another small heart,
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551:. Master Bridge Series. London: Victor Gollancz Ltd in association with Peter Crawley.
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2 back toward hand while RHO follows low. Now there are six vacant places for
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right. At the moment of decision we can perform a vacant places calculation.
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382:); we reached a heart contract without further bidding by the opponents;
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The probabilities of other suit divisions can be calculated similarly.
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13/26 × 12/25 × 13/24 × 12/23 = (3 × 13) / (23 × 25) = 0.067826.
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How Vacant Places calculations work away from the table
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Probability of suit distributions in two hidden hands
374:Alternatively, suppose that LHO dealt and opened 2
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19:"Law of Vacant Places" redirects here. For the
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587:. p. 278.
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1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1711:
1709:
1705:
1704:
1702:
1701:
1696:
1691:
1686:
1680:
1678:
1670:
1669:
1659:
1658:
1655:
1654:
1651:
1650:
1648:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1606:
1604:
1600:
1599:
1597:
1596:
1591:
1582:
1580:Winkle squeeze
1577:
1572:
1567:
1565:Triple squeeze
1562:
1557:
1552:
1547:
1542:
1537:
1535:Simple squeeze
1532:
1527:
1525:Pseudo-squeeze
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1490:Double squeeze
1487:
1482:
1477:
1472:
1467:
1462:
1453:
1448:
1443:
1438:
1433:
1428:
1426:Loser on loser
1423:
1418:
1413:
1411:Crocodile coup
1408:
1403:
1398:
1393:
1388:
1383:
1374:
1369:
1364:
1354:
1352:
1348:
1347:
1345:
1344:
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1242:Avoidance play
1239:
1233:
1231:
1223:
1222:
1212:
1211:
1208:
1207:
1204:
1203:
1201:
1200:
1194:
1192:
1188:
1187:
1185:
1184:
1179:
1174:
1169:
1164:
1159:
1157:Precision Club
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1087:Bidding system
1084:
1079:
1074:
1068:
1066:
1062:
1061:
1059:
1058:
1053:
1048:
1043:
1038:
1033:
1028:
1026:Takeout double
1023:
1018:
1013:
1008:
1003:
998:
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
881:Board (bridge)
878:
876:Bidding system
873:
868:
863:
857:
855:
847:
846:
836:
835:
832:
831:
828:
827:
825:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
697:Bridge scoring
694:
689:
684:
679:
674:
672:Auction bridge
668:
666:
658:
657:
647:
646:
639:
638:
631:
624:
616:
609:
608:
593:
564:
557:
535:
533:
530:
529:
528:
523:
518:
513:
508:
501:
498:
488:
466:
465:
416:
414:
407:
401:
398:
370:
369:
331:
330:
326:
325:
255:
254:
228:
227:
223:
222:
196:
195:
146:
144:
137:
131:
128:
81:
80:
42:
40:
33:
15:
9:
6:
4:
3:
2:
2762:
2751:
2748:
2746:
2743:
2742:
2740:
2717:
2714:
2713:
2711:
2707:
2701:
2698:
2697:
2695:
2691:
2685:
2684:
2680:
2678:
2677:
2673:
2671:
2670:
2666:
2659:
2657:
2654:
2653:
2651:
2647:
2641:
2640:
2636:
2634:
2633:
2629:
2627:
2626:
2622:
2620:
2619:
2615:
2613:
2612:
2608:
2606:
2605:
2601:
2599:
2598:
2594:
2592:
2591:
2587:
2585:
2584:
2580:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2551:
2549:
2546:
2544:
2541:
2540:
2538:
2534:
2530:
2523:
2519:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2369:Mini-Spingold
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2237:
2235:
2231:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2139:Rosenblum Cup
2137:
2135:
2134:McConnell Cup
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2106:
2104:
2100:
2094:
2091:
2090:
2088:
2084:
2080:
2076:Championships
2073:
2069:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1975:
1972:
1971:
1969:
1965:
1959:
1956:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1925:
1923:
1919:
1913:
1910:
1908:
1905:
1903:
1900:
1899:
1897:
1893:
1887:
1884:
1882:
1879:
1877:
1874:
1873:
1871:
1867:
1861:
1860:United States
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1712:
1710:
1706:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1681:
1679:
1675:
1671:
1664:
1660:
1646:
1643:
1641:
1638:
1636:
1635:Rusinow leads
1633:
1631:
1628:
1626:
1625:Rule of 10-12
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1607:
1605:
1603:Defender play
1601:
1595:
1592:
1590:
1583:
1581:
1578:
1576:
1573:
1571:
1570:Trump squeeze
1568:
1566:
1563:
1561:
1560:Strip squeeze
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1505:Guard squeeze
1503:
1501:
1498:
1496:
1495:Entry squeeze
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1475:Clash squeeze
1473:
1471:
1468:
1466:
1463:
1461:
1454:
1452:
1449:
1447:
1444:
1442:
1441:Scissors coup
1439:
1437:
1434:
1432:
1431:Merrimac coup
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1386:Alcatraz coup
1384:
1382:
1375:
1373:
1370:
1368:
1365:
1363:
1356:
1355:
1353:
1351:Declarer play
1349:
1343:
1342:Vacant Places
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1302:Probabilities
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1234:
1232:
1228:
1224:
1217:
1213:
1199:
1196:
1195:
1193:
1189:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1069:
1067:
1063:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1011:Single suiter
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
891:Brown sticker
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
861:Balanced hand
859:
858:
856:
852:
848:
841:
837:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
792:Rubber bridge
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
687:Bridge maxims
685:
683:
682:Bridge-O-Rama
680:
678:
677:Bridge ethics
675:
673:
670:
669:
667:
663:
659:
652:
648:
644:
637:
632:
630:
625:
623:
618:
617:
614:
604:
600:
596:
594:0-943855-48-9
590:
586:
582:
581:
576:
568:
560:
558:0-575-02799-1
554:
550:
546:
540:
536:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
503:
497:
494:
487:
484:
481:
477:
473:
462:
459:
451:
441:
437:
433:
427:
426:
422:
417:This section
415:
411:
406:
405:
397:
385:
381:
333:
332:
328:
327:
289:
288:
285:
283:
278:
269:
230:
229:
225:
224:
202:
201:
192:
189:
181:
171:
167:
163:
157:
156:
152:
147:This section
145:
141:
136:
135:
127:
125:
124:Bayes Theorem
121:
116:
114:
110:
105:
103:
98:
95:
91:
88:
77:
74:
66:
56:
50:
48:
41:
32:
31:
26:
23:episode, see
22:
2693:TV and Radio
2681:
2674:
2667:
2660:
2637:
2630:
2623:
2616:
2609:
2602:
2595:
2588:
2581:
2574:
2552:
2309:Goren Trophy
2264:
2109:Bermuda Bowl
1645:Smith signal
1620:Opening lead
1584:
1575:Vice squeeze
1550:Squeeze play
1455:
1421:Devil's coup
1376:
1357:
1341:
1322:Smother play
1292:Pin (bridge)
1257:Card reading
1172:Säffle Spade
1167:Romex system
1142:Little Major
1051:Weak two bid
1046:Void (cards)
1031:Three suiter
916:Forcing pass
777:Masterpoints
702:Bridge whist
578:
567:
548:
545:Kelsey, Hugh
539:
495:
491:
485:
479:
474:on the page
471:
469:
454:
445:
430:Please help
418:
383:
373:
281:
279:
270:
258:
184:
175:
160:Please help
148:
117:
106:
99:
93:
84:
69:
60:
44:
2240:Buffett Cup
1938:Crockford's
1907:Dallas Aces
1869:Other lists
1850:Switzerland
1820:New Zealand
1755:Netherlands
1451:Vienna coup
1372:Safety play
1191:Conventions
1152:Polish Club
1117:Carrot Club
1097:Boring Club
1021:Strong pass
986:Psychic bid
911:Forcing bid
871:Bidding box
521:Safety play
448:August 2013
178:August 2013
63:August 2022
2739:Categories
2663:Magazines:
2354:Marcus Cup
2154:Venice Cup
1958:Savoy Club
1630:Rule of 11
1446:Trump coup
1162:Roman Club
1056:Zar Points
1036:Two suiter
1016:Stolen bid
956:Minor suit
951:Major suit
782:Minibridge
532:References
2649:Magazines
2429:Reisinger
1912:Four Aces
1902:Blue Team
1715:Australia
1458:Squeezes:
1391:Bath coup
1247:Beer card
1220:Card play
1092:Blue Club
707:Bridgette
419:does not
149:does not
87:card game
2459:Spingold
1830:Pakistan
1735:Bulgaria
1337:Uppercut
1317:Shooting
1127:Fantunes
966:Overcall
921:Game try
655:Overview
603:96188639
500:See also
480:a priori
111:and the
2086:General
1795:Ireland
1785:Hungary
1780:Germany
1770:Fiction
1765:England
1750:Denmark
1730:Britain
1720:Austria
1677:General
1367:Finesse
1282:Hold up
1267:Endplay
1230:General
1147:Moscito
1065:Systems
971:Preempt
901:Cue bid
854:General
844:Bidding
822:Vugraph
752:Goulash
717:Chicago
665:General
440:removed
425:sources
366:
362:
354:
350:
346:
342:
338:
329:
322:
318:
314:
306:
302:
298:
294:
251:
247:
243:
239:
235:
226:
219:
215:
211:
207:
170:removed
155:sources
109:auction
85:In the
2577:Books:
1855:Taiwan
1845:Sweden
1840:Russia
1835:Poland
1825:Norway
1815:Monaco
1810:Mexico
1800:Israel
1775:France
1740:Canada
1725:Brazil
1640:Signal
1379:Coups:
1360:Basic:
1307:Revoke
797:Screen
601:
591:
555:
92:, the
90:bridge
2536:Books
2102:World
1921:Clubs
1895:Teams
1805:Italy
1790:India
1760:Egypt
1745:China
1327:Tempo
1272:Entry
1252:Caddy
817:Trump
282:other
21:Fargo
1401:Coup
1312:Ruff
1262:Duck
1122:EHAA
1082:Acol
807:Suit
599:LCCN
589:ISBN
553:ISBN
423:any
421:cite
380:weak
153:any
151:cite
113:play
102:deal
434:by
384:and
164:by
2741::
597:.
368:x
324:x
253:3
221:2
635:e
628:t
621:v
605:.
561:.
461:)
455:(
450:)
446:(
442:.
428:.
393:♥
389:♥
378:(
376:♠
364:x
359:♠
356:3
352:7
348:8
344:J
340:A
335:♥
320:x
316:x
311:♠
308:2
304:6
300:9
296:K
291:♥
274:♥
265:♥
261:♥
249:7
245:8
241:J
237:A
232:♥
217:6
213:9
209:K
204:♥
191:)
185:(
180:)
176:(
172:.
158:.
76:)
70:(
65:)
61:(
51:.
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
