Main   Study 7Z79 by Richard Pavlicek  

Let’s Make a Deal

Would you trade your wallet for what’s behind Door Number 2? I didn’t think so, but you might be interested in the makeup of a bridge deal. For example, what are the chances of all four hands being balanced? Or a deal having a void suit? Or a seven-card suit? Or various other items of interest — or noninterest for that matter. Check out the following tables for answers.
General CriteriaSuit LengthsFreakness

The number of possible bridge deals is mind-boggling: 53,644,737,765,488,792,839,237,440,000. That’s 29 digits or 53+ octillion. Nobody ever listed or counted each individual deal — such a feat would be impossible even by the most powerful computers — but the number is easily calculated.*

*The formula is 52!/(13!)^4 which means “52 factorial divided by 13 factorial to the fourth power.” The factorial of a number is the product of all integers from 1 through that number. See my article Mapping Bridge Deals for a more detailed explanation of this calculation.

Each table lists a description of various deal types, the number of generic dealprints*, the number of actual deals, the approximate odds against such a deal type, and the percent probability rounded to four decimal places. (In the few cases where the odds are in favor of the deal, it is indicated simply as “favored.”)

*The shape of all four hands. There are 37,478,624 specific dealprints; but all my references are to generic dealprints, which reduce the number to 393,197. Each generic dealprint contains 96, 48, 24 or 16 specific dealprints. See my article Dealprints and Matrices for an explanation.

General Criteria

This table lists deal types based on general characteristics. Some terms may need clarification: “Balanced” includes generic hand patterns 4-3-3-3, 4-4-3-2 and 5-3-3-2; “semibalanced” includes 5-4-2-2 and 6-3-2-2; “quasibalanced” includes 4-4-4-1 and 5-4-3-1. “Symmetrical” means that a like rotation of each specific hand pattern produces the next hand pattern.

Deal TypeDealprintsNumber of Deals — Odds AgainstPercent
No singleton or void46111065866297144810274506240000 — 4:120.6281
At least 1 singleton or void39273642578871468343982564731200000 (favored)79.3719
…in each hand2221531184748234815985942742387200 — 44:12.2085
All hands balanced914651440732749946606831360000 — 10:18.6708
…at least semibalanced42310953740328040921416445440000 — 4:120.4190
…at least quasibalanced128220531630725871382306549360000 — 8:538.2733
All hand patterns same252285229338401870353594581216 — 187:10.5317
…and symmetrical5783265059027355742533946896 — 643:10.1552
Each hand 1 card each rank393197876488338465357824 — 61204166000:10.0000
…and symmetrical5767108864 — 799368884645235431777:10.0000
One side entirely 2 suits28649034882160000 — 82653088824683:10.0000
…and specified28108172480360000 — 495918532948103:10.0000
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Suit Lengths

The following 14 tables show the chances of each specific suit length from a void through 13 cards. Each table caption shows the average number per deal, and the table lists a variety of pertinent holdings. The shorthand “1+ voids” means at least 1 void. Ostensibly missing entries (e.g., 11 voids) are impossible.

Voids — 0.2047
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No void2454743786641333457372963248979200 (favored)81.6234
1 void753908799457894227700441632304128 — 5:116.4032
1+ voids3686509858096432031419875988460800 — 9:218.3766
2 voids113242998506861918196340608870400 — 52:11.8613
2+ voids2932601058638537803719434356156672 — 49:11.9734
3 voids9734658348592829119872339137792 — 918:10.1088
3+ voids18001860131675885523093747286272 — 891:10.1121
4 voids559361760151254006403709124544 — 30476:10.0033
4+ voids826721783083056403221408148480 — 30084:10.0033
5 voids2017822741159103390139399936 — 2358926:10.0000
5+ voids2673622931802396817699023936 — 2339315:10.0000
6 voids5483189905533721240847744 — 282481171:10.0000
6+ voids6558190643293427559624000 — 281388013:10.0000
7 voids792733838125172226048 — 73101595467:10.0000
7+ voids1075737759706318776256 — 72713022011:10.0000
8 voids2583920126156514360 — 13684441679597:10.0000
8+ voids2833921581146550208 — 13679364460601:10.0000
9 voids121454241192768 — 36888473543636524:10.0000
9+ voids251454990035848 — 36869488067815849:10.0000
10 voids12748843056 — 71636823411351487298:10.0000
10+ voids13748843080 — 71636821115431543866:10.0000
12 voids124 — 2235197406895366368301559999:10.0000
Shortest suit 03686509858096432031419875988460800 — 9:218.3766

Singletons — 1.2810
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No singleton2391913436267408464750462079611896 — 3:125.0468
1 singleton7408719692753394437851528571129216 — 5:336.7096
1+ singletons36927840208470357024042377157828104 (favored)74.9532
2 singletons11517113982420636500124063619552240 — 3:126.0649
2+ singletons29519120515716962586190848586698888 — 8:538.2437
3 singletons962445234722137247397227258877376 — 9:19.7581
3+ singletons1800206533296326086066784967146648 — 7:112.1788
4 singletons553001148260368984438039385190616 — 45:12.1405
4+ singletons837761298574188838669557708269272 — 40:12.4207
5 singletons20868138766430185365695589766656 — 385:10.2587
5+ singletons28476150313819854231518323078656 — 355:10.2802
6 singletons627111122755677912332785217536 — 4821:10.0207
6+ singletons760811547389668865822733312000 — 4644:10.0215
7 singletons1036408790415278227335528448 — 131226:10.0008
7+ singletons1337424633990953489948094464 — 126330:10.0008
8 singletons26815665802824130246567936 — 3424320:10.0000
8+ singletons30115843575675262612566016 — 3385897:10.0000
9 singletons22174134142810284507136 — 308065590:10.0000
9+ singletons33177772851132365998080 — 301760011:10.0000
10 singletons103638500218106490880 — 14743640111:10.0000
10+ singletons113638708322081490944 — 14742796897:10.0000
12 singletons1208103975000064 — 257778534818818:10.0000
Shortest suit 12408632720775036312562688742739200 (favored)60.9953

Doubletons — 3.2940
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No doubleton30734804259249659202740664528512 — 65:11.4992
1 doubleton811954243260184157275735384000512 — 11:17.9099
1+ doubletons36246352840478515829590098572911488 (favored)98.5008
2 doubletons11333910928139270707171912961848832 — 4:120.3713
2+ doubletons28126848597218331672314363188910976 (favored)90.5908
3 doubletons8970914440004290394498870749274112 — 8:326.9178
3+ doubletons16792937669079060965142450227062144 (favored)70.2195
4 doubletons5066213058442226453447625951590272 — 3:124.3424
4+ doubletons7822023229074770570643579477788032 — 4:343.3017
5 doubletons196626697224638661855788498511360 — 7:112.4844
5+ doubletons2755810170632544117195953526197760 — 4:118.9592
6 doubletons63482855610546731060215809254400 — 17:15.3232
6+ doubletons78963473407905455340165027686400 — 14:16.4748
7 doubletons1263520926012825773891491584000 — 101:10.9711
7+ doubletons1548617797358724279949218432000 — 85:11.1516
8 doubletons24488573369500118130779008000 — 604:10.1651
8+ doubletons28596871345898506057726848000 — 552:10.1806
9 doubletons326829571339090287856640000 — 7853:10.0127
9+ doubletons418297976398387926947840000 — 6463:10.0155
10 doubletons81454751357497884892160000 — 36874:10.0027
10+ doubletons91468405059297639091200000 — 36531:10.0027
12 doubletons113653701799754199040000 — 3928950:10.0000
Shortest suit 246011015900532747294908106240000 — 4:120.5349

Tripletons — 4.5813
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No tripleton42267298481305036862547953966208 — 178:10.5564
1 tripleton940201641241485067199149856037888 — 31:13.0595
1+ tripletons35093053346256460451930291283473792 (favored)99.4436
2 tripletons1129605355205218885833912073927168 — 9:19.9827
2+ tripletons25691051705014975384731141427435904 (favored)96.3841
3 tripletons792738872474055429629157573119488 — 5:116.5393
3+ tripletons14395046349809756498897229353508736 (favored)86.4014
4 tripletons4182411257865437176662107015225728 — 4:120.9860
4+ tripletons6467737477335701069268071780389248 (favored)69.8621
5 tripletons160929975568326621327582382264320 — 4:118.5956
5+ tripletons2285326219470263892605964765163520 — 1:148.8761
6 tripletons52057474355676760133808619315200 — 6:113.9331
6+ tripletons676116243901937271278382382899200 — 7:330.2805
7 tripletons12485087619301709484239192064000 — 9:19.4839
7+ tripletons15568769546260511144573763584000 — 5:116.3474
8 tripletons2582208867758155132312043520000 — 23:14.1176
8+ tripletons3083681926958801660334571520000 — 13:16.8635
9 tripletons401078084931631556552704000000 — 48:12.0097
9+ tripletons501473059200646528022528000000 — 35:12.7460
10 tripletons9345008504617456103424000000 — 154:10.6431
10+ tripletons10394974269014971469824000000 — 134:10.7363
12 tripletons149965764397515366400000000 — 1072:10.0931
Shortest suit 3149965764397515366400000000 — 1072:10.0931

4-Card Suits — 3.8177
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 4-card suit58548827527196765622468298731000 — 63:11.5426
1 4-card suit1122133264522488820600040694232000 — 15:16.0854
1+ 4-card suits33464952817210568723170370938709000 (favored)98.4574
2 4-card suits1105898107219615979748529508958000 — 11:215.1128
2+ 4-card suits22243649552688079902570330244477000 (favored)92.3719
3 4-card suits6571611328128521386758622862464000 — 4:121.1169
3+ 4-card suits11184741445468463922821800735519000 (favored)77.2592
4 4-card suits3108812122462049388925454412335000 — 7:222.5977
4+ 4-card suits4613130117339942536063177873055000 (favored)56.1422
5 4-card suits105849405039628729954030309488000 — 9:217.5321
5+ 4-card suits1504317994877893147137723460720000 — 2:133.5445
6 4-card suits34025219103350668664679414192000 — 9:19.7290
6+ 4-card suits44598589838264417183693151232000 — 5:116.0125
7 4-card suits8392635160299611246929602080000 — 19:14.9122
7+ 4-card suits10573370734913748519013737040000 — 14:16.2834
8 4-card suits181585877783479958092483360000 — 90:11.0921
8+ 4-card suits218735574614137272084134960000 — 71:11.3712
9 4-card suits29124079826585336167145600000 — 431:10.2313
9+ 4-card suits37149696830657313991651600000 — 357:10.2791
10 4-card suits724628911563140240356000000 — 2177:10.0459
10+ 4-card suits825617004071977824506000000 — 2093:10.0478
12 4-card suits1988092508837584150000000 — 54290:10.0018
Longest suit 4291571494042604960223750000000 — 33:12.9294
Longest 4 shortest 19226907515517134602150000000 — 235:10.4230
Longest 4 shortest 2191294620762690310255200000000 — 40:12.4133
Longest 4 shortest 3149965764397515366400000000 — 1072:10.0931

5-Card Suits — 1.9951
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 5-card suit870064597881752913411186598943736 — 10:18.5710
1 5-card suit12114314162753528304985663093951488 — 3:126.4010
1+ 5-card suits30619149046856012575381652638496264 (favored)91.4290
2 5-card suits10450818067215541367691851979855600 — 2:133.6794
2+ 5-card suits18504834884102484270395989544544776 (favored)65.0280
3 5-card suits5479811640282689397513831787179072 — 7:221.6988
3+ 5-card suits8054016816886942902704137564689176 — 9:431.3486
4 5-card suits200554206420083748200765168830680 — 11:17.8413
4+ 5-card suits257425176604253505190305777510104 — 9:19.6498
5 5-card suits4722847666043311209636417007104 — 62:11.5801
5+ 5-card suits5687970184169756989540608679424 — 54:11.8085
6 5-card suits878115132376736925239588728832 — 464:10.2146
6+ 5-card suits965122518126445779904191672320 — 436:10.2284
7 5-card suits756872885114033959677296640 — 7804:10.0128
7+ 5-card suits877385749708854664602943488 — 7262:10.0138
8 5-card suits12512864594820704925646848 — 104597:10.0010
Longest suit 5329721583251210971361009130800768 — 3:240.2337
Longest 5 shortest 015881609503787729626328278490752 — 32:13.0003
Longest 5 shortest 1149112670231342665127083546710016 — 3:123.6188
Longest 5 shortest 22187303516080576607597305600000 — 6:113.6146

6-Card Suits — 0.6650
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 6-card suit11385326905653785695390696041730560 (favored)50.1553
1 6-card suit14462119066726906544846134962690048 — 7:435.5426
1+ 6-card suits27934426739083979793402143195709440 — 1:149.8447
2 6-card suits933536485428597816725021394136064 — 7:112.0896
2+ 6-card suits1347237672357073248556008233019392 — 6:114.3022
3 6-card suits326841111477008094705239177535488 — 47:12.0719
3+ 6-card suits413701186928475431830986838883328 — 44:12.2126
4 6-card suits742374270503985138725291822080 — 721:10.1384
4+ 6-card suits868675451467337125747661347840 — 709:10.1407
5 6-card suits10921166137252500210225371136 — 46001:10.0022
5+ 6-card suits12631180963351987022369525760 — 45423:10.0022
6 6-card suits15414747938755361745117184 — 3637438:10.0000
6+ 6-card suits17114826099486812144154624 — 3618262:10.0000
7 6-card suits1273663812654622654464 — 728237322:10.0000
7+ 6-card suits1778160731450399037440 — 686338736:10.0000
8 6-card suits54496918795776382976 — 11929220917:10.0000
Longest suit 63287322642122348654241172787919872 — 4:342.2075
Longest 6 shortest 0256115023219452210248132366595072 — 9:19.3639
Longest 6 shortest 1707715313265176067504842881484800 — 5:228.5457
Longest 6 shortest 21852305637720376488197539840000 — 22:14.2980
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7-Card Suits — 0.1411
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 7-card suit14861246668614955529607643028925952 (favored)86.9957
1 7-card suit1666306414841838510523780739153920 — 7:111.9580
1+ 7-card suits2445856976122809959185196208514048 — 13:213.0043
2 7-card suits67357531829247712684682606230528 — 99:10.9914
2+ 7-card suits77955561280971448661415469360128 — 94:11.0463
3 7-card suits1013628879408411913023879090176 — 1856:10.0538
3+ 7-card suits1059829451723735976732863129600 — 1820:10.0549
4 7-card suits462572315324063708984039424 — 93731:10.0011
Longest suit 7848376782984599117957218132857856 — 13:212.6443
Longest 7 shortest 0765322601365526899294160773154816 — 19:14.8492
Longest 7 shortest 182674069493103114774199298903040 — 12:17.5860
Longest 7 shortest 238112125969103888858060800000 — 477:10.2090

8-Card Suits — 0.0187
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 8-card suit21802052659418358646318763250088120 (favored)98.1633
1 8-card suit135330969336127795929767968558464 — 54:11.8070
1+ 8-card suits175177985319406842474075987351880 — 53:11.8367
2 8-card suits3533215717541316474557292763888 — 3412:10.0293
2+ 8-card suits3984715983279046544308018793416 — 3355:10.0298
3 8-card suits4277262832169584598962242752 — 204101:10.0005
3+ 8-card suits4515265737730069750726029528 — 201869:10.0005
4 8-card suits2382905560485151763786776 — 18462783:10.0000
Longest suit 8101924981975954511555218232092504 — 53:11.8305
Longest 8 shortest 096759559206840312485595147564504 — 94:11.0424
Longest 8 shortest 15165422769114199069623084528000 — 125:10.7881

9-Card Suits — 0.0015
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 9-card suit28224053565418928084244786753267000 (favored)99.8521
1 9-card suit9375679148608957994535139752000 — 676:10.1475
1+ 9-card suits11095779318837404548052484173000 — 675:10.1479
2 9-card suits15568169510962571319297214000 — 316466:10.0003
2+ 9-card suits17201170228446553517344421000 — 315132:10.0003
3 9-card suits1521712788941101915392000 — 75260338:10.0000
3+ 9-card suits1633717483982198047207000 — 74767853:10.0000
4 9-card suits1124695041096131815000 — 11425829223:10.0000
Longest suit 98484479298562143148725113050600 — 675:10.1478
Longest 9 shortest 08302561390081488049420251922920 — 872:10.1144
Longest 9 shortest 1181917908480655099304861127680 — 2994:10.0334

10-Card Suits — 0.0001
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 10-card suit33359353641205490163369327919048832 (favored)99.9934
1 10-card suit536773531671663756659651717632 — 15188:10.0066
1+ 10-card suits596043532275325423511318391168 — 15186:10.0066
2 10-card suits5447603182972886040115712 — 88936093:10.0000
2+ 10-card suits5927603661666851666673536 — 88865568:10.0000
3 10-card suits433476472744726730752 — 112587211668:10.0000
3+ 10-card suits480478693965626557824 — 112064787979:10.0000
4 10-card suits472221220899827072 — 24151014322647:10.0000
Longest suit 10528733532238785856985879197952 — 15186:10.0066
Longest 10 shortest 0526153331934692003952959212288 — 16099:10.0062
Longest 10 shortest 1258200304093853032919985664 — 267815:10.0004

11-Card Suits — 0.0000
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 11-card suit36780753644659643413038984236251264 (favored)99.9999
1 11-card suit2390178121509765066018800640 — 686682:10.0001
1+ 11-card suits2539078122075753855001188736 — 686677:10.0001
2 11-card suits1385565933048294288896 — 94789901255:10.0000
2+ 11-card suits1489565988788982388096 — 94780565992:10.0000
3 11-card suits8755476844780032 — 966975284520818:10.0000
3+ 11-card suits10455740688099200 — 962398197704679:10.0000
4 11-card suits17263843319168 — 203320432500058718:10.0000
Longest suit 112431978122061820624147552512 — 686677:10.0001
Longest 11 shortest 02431978122061820624147552512 — 686677:10.0001

12-Card Suits — 0.0000
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 12-card suit38586153644737080202550239380997624 (favored)100.0000
1 12-card suit7113685286153633192746752 — 78280783:10.0000
1+ 12-card suits7336685286242599856442376 — 78280773:10.0000
2 12-card suits20988966097766000 — 602979551902859:10.0000
2+ 12-card suits22388966663695624 — 602975716264015:10.0000
3 12-card suits9559760448 — 95835170128863397005:10.0000
3+ 12-card suits14565929624 — 94790474805554255344:10.0000
4 12-card suits56169176 — 8695608257162511304464:10.0000
Longest suit 127269685286242093646948664 — 78280773:10.0000
Longest 12 shortest 07269685286242093646948664 — 78280773:10.0000

13-Card Suits — 0.0000
Deal ContainsDealprintsNumber of Deals — Odds AgainstPercent
No 13-card suit39226553644737764137143270820420728 (favored)100.0000
1 13-card suit9191351649567668176192 — 39688347517:10.0000
1+ 13-card suits9321351649568417019272 — 39688347495:10.0000
2 13-card suits12748843056 — 71636823411351487298:10.0000
2+ 13-card suits13748843080 — 71636821115431543866:10.0000
4 13-card suits124 — 2235197406895366368301559999:10.0000
Longest suit 139321351649568417019272 — 39688347495:10.0000
Longest 13 shortest 09321351649568417019272 — 39688347495:10.0000
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Freakness

The flattest deal (each hand 4-3-3-3) has a freakness of zero, and the wildest deal (each hand 13-0-0-0) has a freakness of 80. All others fall in between, with the great majority on the low end. The average deal freakness is 11.9314. See Patterns and Freakness for an explanation. The following table shows a breakdown of deals by freakness (missing freakness numbers are impossible).

FreaknessDealprintsNumber of Deals — Odds AgainstPercent
0149965764397515366400000000 — 1072:10.0931
13224845939788819148800000000 — 237:10.4191
24337268909683228723200000000 — 158:10.6287
3171277249852411486553600000000 — 41:12.3809
4191059656755610994244704000000 — 49:11.9753
5542182129845650489839104000000 — 23:14.0677
6612100679403961990102451200000 — 24:13.9159
71443900852209396223412531200000 — 12:17.2716
81692935429792437846953208960000 — 17:15.4720
92773902968973315092438517760000 — 12:17.2756
104494319233756510792955996160000 — 11:18.0516
115924177391033763705615137280000 — 11:17.7871
128533990368759941243838222064000 — 12:17.4385
1312063836417947826698802760192000 — 12:17.1515
1416013648437681407219980639360000 — 13:16.8011
1522263264937827762008134953216000 — 15:16.0862
1627492663243649725330822345395200 — 19:14.9646
1735432289468162846201949464652800 — 22:14.2678
1846081932541204471813293546988800 — 26:13.6025
1953911480279600703791494954547200 — 35:12.7594
2066851174583424431230888775500800 — 44:12.1896
217689856279205971843936824622080 — 61:11.5962
229282651189608460944685643230720 — 81:11.2139
2310881468514461078731756738649600 — 113:10.8734
2411530309869707607739903405614976 — 172:10.5776
2513463224014273521217767999290880 — 238:10.4176
2614828146819296583833277172307968 — 364:10.2737
271594393044009812336198278474240 — 575:10.1734
281724660185626811626055142978432 — 890:10.1122
291706435602130280545690996040192 — 1505:10.0664
301882822690215766463274527427072 — 2363:10.0423
311893012814634946132905421279744 — 4185:10.0239
32188117271032073391375174407808 — 7376:10.0136
33189604027998015429852085509888 — 13316:10.0075
34178242134460884602256692932736 — 25131:10.0040
35182641204090604570608544934144 — 44551:10.0022
3616955588171565475675378600832 — 91204:10.0011
3715676281495292714510093483264 — 190569:10.0005
3814636136548415355635932352704 — 392861:10.0003
391279666675424468034146976640 — 804564:10.0001
401224431568800006311999095808 — 1699295:10.0001
411029612551152845439469866176 — 4274087:10.0000
4287945539188505944246918336 — 9684583:10.0000
4374942281818876604493219136 — 23509637:10.0000
4466081080773542988045704512 — 49635501:10.0000
455400364042919909640464064 — 147358276:10.0000
464335149341040837809172352 — 359209613:10.0000
47366456213864818077979584 — 954297269:10.0000
48292722667027728510788480 — 2366641907:10.0000
4924057918128597821105664 — 6774926309:10.0000
5019212786029111818995328 — 19254909266:10.0000
5115451050155184192903360 — 51082676705:10.0000
521176381937032300814848 — 140454402763:10.0000
53914123106992036215616 — 435757034414:10.0000
5483248759791163887232 — 1100183911476:10.0000
5550012332598988387392 — 4349832327799:10.0000
564956378802488829056 — 8409844615729:10.0000
572881201770056710080 — 44638104823765:10.0000
58350776728909674432 — 69064942861433:10.0000
59126101120237302656 — 530504468704205:10.0000
6020787806733670656 — 610941046579621:10.0000
61567741853982720 — 6929184906512716:10.0000
621429610252313376 — 5582032189812906:10.0000
6321384475703040 — 139526990499859266:10.0000
6487927265749696 — 57852603509917176:10.0000
65613029299712 — 4117238758126189063:10.0000
664877879337120 — 688818623132738764:10.0000
671361924992 — 148220595292542806325:10.0000
68305262221952 — 10194313021156427427:10.0000
7012288527616 — 185925834445908958811:10.0000
721014638104 — 3664732657008639427566:10.0000
742421824 — 127173270761001727827807:10.0000
76224336 — 2204336693190696615682010:10.0000
80124 — 2235197406895366368301559999:10.0000
Totals39319753644737765488792839237440000100.0000
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Photo is Monty Hall of the old TV show Let’s Make a Deal.
© 2004 Richard Pavlicek