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Ask the Wizard #317

Golf Digest claims that if four amateur golfers play a round that the probability that two get a hole-in-one on the same hole are 1 in 26,000,000. Do you agree with that figure?

DHass22

No, I don't.

I will have to take on faith their statistic that for an amateur golfer, the probability of a hole-in-one are 1 in 12,500 per hole.

However, almost all hole-in-ones are achieved on par-3 holes. The typical course will have four such holes. So, the probability of a hole-in-one on a par 3 hole is (1/12500) × (4/18) = 1 in 2,778.

That said, the probability that exactly two golfers out of four get a hole-in-one on a par 3 hole is combin(4,2) × (1/12500)2 × (12449/12500)2 = 1 in 1,286,935.

Assuming four par-3 holes in a round, the probability of two hole-in-ones in the same hole are 4 × (1/1,286,935) = 1 in 321,734.

Where Golf Digest seems to have gone wrong is in assuming every hole has the same probability of a hole in one.

This question is raised and discussed in my forum at Wizard of Vegas.

A number of sports books, like the MGM in New Jersey, are offering a truly free bet — if the bet looses, they refund in CASH you can then withdraw. Like a 100% loss rebate for slots, but applied to sports betting. What kind of bet should I make under this promotion?

"Anonymous" .

With any free-roll like this, you should shoot for the moon. If they allow parlays, then parlay bet as much as they allow under this promotion and parlay enough events to get up to whatever their maximum payout is, which in Las Vegas is typically around $100,000.

If you have to bet on a single event, then find the biggest long-shot you can.

As an example of why this strategy is best, consider the same promotion in double-zero roulette with a $100 bet. If you make an even-money bet, then the expected amount you'll get back is (18/38)×$200 + (20/38)×$100 = $147.37. However, if you bet on a single number, then you can expect to get back (1/38)×$3600 + (37/38)×$100 = $192.11.

I found a blackjack dealer who exposes his hole card when peeking for blackjack. He offers insurance after peeking. So, when the dealer has an ace up and I can see a 10, I take insurance. How much does this reduce the house edge of the overall game?

"Anonymous" .

Let's assume six decks. The probability of the up card being an ace is 1/13. Next, the probability the hole card is a 10-point card is 96/311. So, this opportunity will occur 2.37% of hands.

The insurance bet I assume will be half a unit and insurance pays 2 to 1. So, every time this happens, you can expect to gain a unit. Since it happens with probability 2.37% of the time, that is what it is worth to you. Subtract the house edge under whatever the blackjack rules are to get the overall player advantage of the game.

Let's say a dealer has chips of every denomination evenly divisible by $1. You buy in at the table for $10. How many different ways can the dealer give change for the $10? How about for other buy-in amounts?

"Anonymous" .

There are 42 ways the dealer could break down $10. Here they are:

10
9,1
8,2
8,1,1
7,3
7,2,1
7,1,1,1
6,4
6,3,1
6,2,2
6,2,1,1
6,1,1,1,1
5,5
5,4,1
5,3,2
5,3,1,1
5,2,2,1
5,2,1,1,1
5,1,1,1,1,1
4,4,2
4,4,1,1
4,3,3
4,3,2,1
4,3,1,1,1
4,2,2,2
4,2,2,1,1
4,2,1,1,1,1
4,1,1,1,1,1,1
3,3,3,1
3,3,2,2
3,3,2,1,1
3,3,1,1,1,1
3,2,2,2,1
3,2,2,1,1,1
3,2,1,1,1,1,1
3,1,1,1,1,1,1,1
2,2,2,2,2
2,2,2,2,1,1
2,2,2,1,1,1,1
2,2,1,1,1,1,1,1
2,1,1,1,1,1,1,1,1
1,1,1,1,1,1,1,1,1,1

Mathematicians call these partitions. Here are the number of partitions for starting quantities up to 405, which is as high as my computer can calculate (2^64).

Common Birthday to 2+ People

Initial
Quantity
Total
Partitions
11
22
33
45
57
611
715
822
930
1042
1156
1277
13101
14135
15176
16231
17297
18385
19490
20627
21792
221002
231255
241575
251958
262436
273010
283718
294565
305604
316842
328349
3310143
3412310
3514883
3617977
3721637
3826015
3931185
4037338
4144583
4253174
4363261
4475175
4589134
46105558
47124754
48147273
49173525
50204226
51239943
52281589
53329931
54386155
55451276
56526823
57614154
58715220
59831820
60966467
611121505
621300156
631505499
641741630
652012558
662323520
672679689
683087735
693554345
704087968
714697205
725392783
736185689
747089500
758118264
769289091
7710619863
7812132164
7913848650
8015796476
8118004327
8220506255
8323338469
8426543660
8530167357
8634262962
8738887673
8844108109
8949995925
9056634173
9164112359
9272533807
9382010177
9492669720
95104651419
96118114304
97133230930
98150198136
99169229875
100190569292
101214481126
102241265379
103271248950
104304801365
105342325709
106384276336
107431149389
108483502844
109541946240
110607163746
111679903203
112761002156
113851376628
114952050665
1151064144451
1161188908248
1171327710076
1181482074143
1191653668665
1201844349560
1212056148051
1222291320912
1232552338241
1242841940500
1253163127352
1263519222692
1273913864295
1284351078600
1294835271870
1305371315400
1315964539504
1326620830889
1337346629512
1348149040695
1359035836076
13610015581680
13711097645016
13812292341831
13913610949895
14015065878135
14116670689208
14218440293320
14320390982757
14422540654445
14524908858009
14627517052599
14730388671978
14833549419497
14937027355200
15040853235313
15145060624582
15249686288421
15354770336324
15460356673280
15566493182097
15673232243759
15780630964769
15888751778802
15997662728555
160107438159466
161118159068427
162129913904637
163142798995930
164156919475295
165172389800255
166189334822579
167207890420102
168228204732751
169250438925115
170274768617130
171301384802048
172330495499613
173362326859895
174397125074750
175435157697830
176476715857290
177522115831195
178571701605655
179625846753120
180684957390936
181749474411781
182819876908323
183896684817527
184980462880430
1851071823774337
1861171432692373
1871280011042268
1881398341745571
1891527273599625
1901667727404093
1911820701100652
1921987276856363
1932168627105469
1942366022741845
1952580840212973
1962814570987591
1973068829878530
1983345365983698
1993646072432125
2003972999029388
2014328363658647
2024714566886083
2035134205287973
2045590088317495
2056085253859260
2066622987708040
2077206841706490
2087840656226137
2098528581302375
2109275102575355
21110085065885767
21210963707205259
21311916681236278
21412950095925895
21514070545699287
21615285151248481
21716601598107914
21818028182516671
21919573856161145
22021248279009367
22123061871173849
22225025873760111
22327152408925615
22429454549941750
22531946390696157
22634643126322519
22737561133582570
22840718063627362
22944132934884255
23047826239745920
23151820051838712
23256138148670947
23360806135438329
23465851585970275
23571304185514919
23677195892663512
23783561103925871
23890436839668817
23997862933703585
240105882246722733
241114540884553038
242123888443077259
243133978259344888
244144867692496445
245156618412527946
246169296722391554
247182973889854026
248197726516681672
249213636919820625
250230793554364681
251249291451168559
252269232701252579
253290726957916112
254313891991306665
255338854264248680
256365749566870782
257394723676655357
258425933084409356
259459545750448675
260495741934760846
261534715062908609
262576672674947168
263621837416509615
264670448123060170
265722760953690372
266779050629562167
267839611730366814
268904760108316360
269974834369944625
2701050197489931117
2711131238503938606
2721218374349844333
2731312051800816215
2741412749565173450
2751520980492851175
2761637293969337171
2771762278433057269
2781896564103591584
2792040825852575075
2802195786311682516
2812362219145337711
2822540952590045698
2832732873183547535
2842938929793929555
2853160137867148997
2863397584011986773
2873652430836071053
2883925922161489422
2894219388528587095
2904534253126900886
2914872038056472084
2925234371069753672
2935622992691950605
2946039763882095515
2956486674127079088
2966965850144195831
2977479565078510584
2988030248384943040
2998620496275465025
3009253082936723602
3019930972392403501
30210657331232548839
30311435542077822104
30412269218019229465
30513162217895057704
30614118662665280005
30715142952738857194
30816239786535829663
30917414180133147295
31018671488299600364
31120017426762576945
31221458096037352891
31323000006655487337
31424650106150830490
31526415807633566326
31628305020340996003
31730326181989842964
31832488293351466654
31934800954869440830
32037274405776748077
32139919565526999991
32242748078035954696
32345772358543578028
32449005643635237875
32552462044228828641
32656156602112874289
32760105349839666544
32864325374609114550
32968834885946073850
33073653287861850339
33178801255302666615
33284300815636225119
33390175434980549623
33496450110192202760
335103151466321735325
336110307860425292772
337117949491546113972
338126108517833796355
339134819180623301520
340144117936527873832
341154043597379576030
342164637479165761044
343175943559810422753
344188008647052292980
345200882556287683159
346214618299743286299
347229272286871217150
348244904537455382406
349261578907351144125
350279363328483702152
351298330063062758076
352318555973788329084
353340122810048577428
354363117512048110005
355387632532919029223
356413766180933342362
357441622981929358437
358471314064268398780
359502957566506000020
360536679070310691121
361572612058898037559
362610898403751884101
363651688879997206959
364695143713458946040
365741433159884081684
366790738119649411319
367843250788562528427
368899175348396088349
369958728697912338045
3701022141228367345362
3711089657644424399782
3721161537834849962850
3731238057794119125085
3741319510599727473500
3751406207446561484054
3761498478743590581081
3771596675274490756791
3781701169427975813525
3791812356499739472950
3801930656072350465812
3812056513475336633805
3822190401332423765131
3832332821198543892336
3842484305294265418180
3852645418340688763701
3862816759503217942792
3872998964447736452194
3883192707518433532826
3893398704041358160275
3903617712763867604423
3913850538434667429186
3924098034535626594791
3934361106170762284114
3944640713124699623515
3954937873096788191655
3965253665124416975163
3975589233202595404488
3985945790114707874597
3996324621482504294325
4006727090051741041926
4017154640222653942321
4027608802843339879269
4038091200276484465581
4048603551759348655060
4059147679068859117602

This question is raised and discussed in my forum at Wizard of Vegas.