I played the old game Risk and thought that at table with probabilities would be nice to have sometimes. So I made one.
Below are the probabilities of winning if you over and over attack the same country and from the beginning you have i troops and the opponent has j troops. The table supposes you always attack with the maximum number of troops you can attack with (=min(3,number of troops you have)), I am quite sure that strategy is optimal. The numbers of troops you attack with are in the rows, the number of troops the opponent uses in the columns. They might be totally wrong, I dunno, they look quite reasonable. I made them up with a simple program I wrote in ten minutes.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 41.66667% | 10.60957% | 2.70151% | 0.68788% | 0.17516% | 0.04460% | 0.01136% | 0.00289% | 0.00074% | 0.00019% | 0.00005% | 0.00001% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% |
| 2 | 75.42438% | 36.26543% | 20.60664% | 9.13035% | 4.91348% | 2.13505% | 1.13288% | 0.48967% | 0.25881% | 0.11170% | 0.05897% | 0.02544% | 0.01343% | 0.00579% | 0.00306% | 0.00132% | 0.00070% | 0.00030% | 0.00016% | 0.00007% | 0.00004% | 0.00002% | 0.00001% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% | 0.00000% |
| 3 | 91.63746% | 65.59540% | 47.02511% | 31.49943% | 20.59419% | 13.36985% | 8.37418% | 5.35023% | 3.27695% | 2.07540% | 1.25542% | 0.79114% | 0.47513% | 0.29854% | 0.17853% | 0.11198% | 0.06679% | 0.04185% | 0.02493% | 0.01561% | 0.00929% | 0.00581% | 0.00346% | 0.00216% | 0.00129% | 0.00080% | 0.00048% | 0.00030% | 0.00018% |
| 4 | 97.15441% | 78.54543% | 64.16229% | 47.65309% | 35.86062% | 25.25027% | 18.14856% | 12.33954% | 8.61723% | 5.71907% | 3.91677% | 2.55451% | 1.72527% | 1.11063% | 0.74234% | 0.47311% | 0.31370% | 0.19835% | 0.13069% | 0.08211% | 0.05382% | 0.03364% | 0.02196% | 0.01366% | 0.00889% | 0.00551% | 0.00357% | 0.00221% | 0.00143% |
| 5 | 99.03171% | 88.97887% | 76.93749% | 63.82946% | 50.62028% | 39.67536% | 29.74181% | 22.40476% | 16.15579% | 11.82753% | 8.29203% | 5.94240% | 4.07854% | 2.87518% | 1.94097% | 1.35060% | 0.89977% | 0.61954% | 0.40830% | 0.27870% | 0.18203% | 0.12336% | 0.07996% | 0.05385% | 0.03468% | 0.02323% | 0.01488% | 0.00992% | 0.00632% |
| 6 | 99.67051% | 93.39797% | 85.69198% | 74.48749% | 63.77205% | 52.06826% | 42.33302% | 32.94825% | 25.77745% | 19.34338% | 14.69768% | 10.72073% | 7.96256% | 5.67884% | 4.14194% | 2.90072% | 2.08466% | 1.43823% | 1.02104% | 0.69565% | 0.48880% | 0.32951% | 0.22951% | 0.15331% | 0.10598% | 0.07024% | 0.04824% | 0.03175% | 0.02168% |
| 7 | 99.88788% | 96.66495% | 90.99420% | 83.37390% | 73.63956% | 64.00731% | 53.55339% | 44.55806% | 35.69336% | 28.67610% | 22.18670% | 17.33135% | 13.03884% | 9.95613% | 7.32065% | 5.48617% | 3.95801% | 2.92021% | 2.07340% | 1.50969% | 1.05743% | 0.76131% | 0.52703% | 0.37576% | 0.25750% | 0.18204% | 0.12364% | 0.08676% | 0.05846% |
| 8 | 99.96185% | 98.03082% | 94.67994% | 88.78009% | 81.84101% | 72.95558% | 64.29414% | 54.73596% | 46.39852% | 37.98720% | 31.17313% | 24.70449% | 19.73474% | 15.22116% | 11.88937% | 8.96380% | 6.87079% | 5.08124% | 3.83283% | 2.78820% | 2.07442% | 1.48771% | 1.09375% | 0.77474% | 0.56368% | 0.39495% | 0.28473% | 0.19759% | 0.14129% |
| 9 | 99.98702% | 99.01146% | 96.69917% | 92.98201% | 87.29365% | 80.76408% | 72.60798% | 64.64118% | 55.80698% | 47.99353% | 39.98733% | 33.37489% | 26.97147% | 21.94329% | 17.27681% | 13.75261% | 10.58886% | 8.27265% | 6.24819% | 4.80325% | 3.56776% | 2.70443% | 1.97971% | 1.48231% | 1.07125% | 0.79344% | 0.56692% | 0.41587% | 0.29415% |
| 10 | 99.99558% | 99.41952% | 98.10996% | 95.39335% | 91.62835% | 86.10911% | 79.98330% | 72.39700% | 65.00598% | 56.75929% | 49.39522% | 41.74948% | 35.33829% | 29.02606% | 23.98018% | 19.21139% | 15.54036% | 12.18213% | 9.67480% | 7.44129% | 5.81543% | 4.39883% | 3.38942% | 2.52626% | 1.92234% | 1.41415% | 1.06417% | 0.77374% | 0.57649% |
| 11 | 99.99850% | 99.70930% | 98.83882% | 97.20410% | 94.30417% | 90.52209% | 85.20487% | 79.41151% | 72.30344% | 65.38261% | 57.62945% | 50.65002% | 43.32782% | 37.11006% | 30.90396% | 25.86774% | 21.03434% | 17.25231% | 13.73603% | 11.06577% | 8.64751% | 6.85658% | 5.27012% | 4.12006% | 3.12035% | 2.40886% | 1.80040% | 1.37426% | 1.01499% |
| 12 | 99.99949% | 99.82967% | 99.34924% | 98.19904% | 96.37007% | 93.35412% | 89.61223% | 84.48643% | 78.98809% | 72.28362% | 65.76179% | 58.42981% | 51.78670% | 44.75635% | 38.72336% | 32.63143% | 27.62416% | 22.75461% | 18.89014% | 15.24621% | 12.43767% | 9.85694% | 7.91645% | 6.17208% | 4.88803% | 3.75536% | 2.93685% | 2.22661% | 1.72160% |
| 13 | 99.99983% | 99.91478% | 99.60302% | 98.93183% | 97.58133% | 95.61124% | 92.54099% | 88.85741% | 83.91562% | 78.67557% | 72.31940% | 66.14007% | 59.17362% | 52.82732% | 46.06187% | 40.20406% | 34.23001% | 29.26510% | 24.38098% | 20.45691% | 16.71064% | 13.78523% | 11.06221% | 8.98692% | 7.09669% | 5.68609% | 4.42523% | 3.50146% | 2.68926% |
| 14 | 99.99994% | 99.95011% | 99.78097% | 99.32126% | 98.49793% | 96.99124% | 94.92899% | 91.83846% | 88.22651% | 83.45706% | 78.44710% | 72.39515% | 66.51474% | 59.86947% | 53.78798% | 47.26425% | 41.57215% | 35.71695% | 30.80373% | 25.92153% | 21.95621% | 18.12876% | 15.10502% | 12.25785% | 10.06155% | 8.03731% | 6.50792% | 5.12444% | 4.09810% |
| 15 | 99.99998% | 99.97505% | 99.86710% | 99.60467% | 99.01508% | 98.06471% | 96.44142% | 94.31805% | 91.23122% | 87.69618% | 83.08786% | 78.28395% | 72.50098% | 66.88436% | 60.52441% | 54.68113% | 48.37909% | 42.84355% | 37.10638% | 32.25127% | 27.38365% | 23.39187% | 19.50096% | 16.39493% | 13.43981% | 11.13527% | 8.98841% | 7.34811% | 5.84806% |
| 16 | 99.99999% | 99.98540% | 99.92754% | 99.75137% | 99.40087% | 98.69662% | 97.64389% | 95.93290% | 93.77194% | 90.70417% | 87.24830% | 82.79018% | 78.17233% | 72.62947% | 67.24804% | 61.14385% | 55.51645% | 49.41867% | 44.03106% | 38.41001% | 33.61731% | 28.77399% | 24.76776% | 20.82825% | 17.65381% | 14.60518% | 12.20412% | 9.94545% | 8.20203% |
| 17 | 100.00000% | 99.99270% | 99.95621% | 99.85729% | 99.61209% | 99.17959% | 98.37706% | 97.24161% | 95.46573% | 93.28390% | 90.24556% | 86.86880% | 82.55074% | 78.10194% | 72.77523% | 67.60530% | 61.73218% | 56.30168% | 50.39290% | 45.14514% | 39.63758% | 34.91008% | 30.09853% | 26.08760% | 22.11200% | 18.88118% | 15.75186% | 13.26498% | 10.90467% |
| 18 | 100.00000% | 99.99573% | 99.97635% | 99.91097% | 99.76801% | 99.45533% | 98.94898% | 98.06273% | 96.86106% | 95.03785% | 92.84756% | 89.84544% | 86.54654% | 82.35923% | 78.06494% | 72.93423% | 67.95589% | 62.29297% | 57.04306% | 51.30984% | 46.19442% | 40.79732% | 36.13667% | 31.36263% | 27.35495% | 23.35381% | 20.07702% | 16.87841% | 14.31547% |
| 19 | 100.00000% | 99.99786% | 99.98575% | 99.94951% | 99.85132% | 99.66331% | 99.28704% | 98.71495% | 97.75796% | 96.50351% | 94.64673% | 92.45713% | 89.49568% | 86.27264% | 82.20758% | 78.05524% | 73.10338% | 68.29974% | 62.82916% | 57.74574% | 52.17615% | 47.18612% | 41.89623% | 37.30324% | 32.57108% | 28.57315% | 24.55539% | 21.24169% | 17.98389% |
| 20 | 100.00000% | 99.99875% | 99.99236% | 99.96870% | 99.91234% | 99.77909% | 99.54687% | 99.11172% | 98.48175% | 97.46528% | 96.16913% | 94.28956% | 92.10747% | 89.18947% | 86.03989% | 82.08937% | 78.06805% | 73.28031% | 68.63687% | 63.34322% | 58.41396% | 52.99740% | 48.12631% | 42.94027% | 38.41517% | 33.72818% | 29.74531% | 25.71849% | 22.37573% |
| 21 | 100.00000% | 99.99937% | 99.99541% | 99.98243% | 99.94429% | 99.86553% | 99.69653% | 99.42191% | 98.93300% | 98.25240% | 97.18615% | 95.85738% | 93.96357% | 91.79407% | 88.92112% | 85.84242% | 81.99943% | 78.09956% | 73.46317% | 68.96735% | 63.83725% | 59.05130% | 53.77826% | 49.02018% | 43.93460% | 39.47714% | 34.83781% | 30.87433% | 26.84485% |
| 22 | 100.00000% | 99.99963% | 99.99756% | 99.98917% | 99.96755% | 99.91264% | 99.81022% | 99.60575% | 99.29117% | 98.75363% | 98.02897% | 96.92127% | 95.56737% | 93.66607% | 91.51296% | 88.68587% | 85.67539% | 81.93356% | 78.14670% | 73.65048% | 69.29134% | 64.31302% | 59.66079% | 54.52273% | 49.87216% | 44.88369% | 40.49330% | 35.90341% | 31.96287% |
| 23 | 100.00000% | 99.99982% | 99.99853% | 99.99397% | 99.97952% | 99.94752% | 99.87427% | 99.74764% | 99.50872% | 99.15691% | 98.57567% | 97.81284% | 96.67081% | 95.29797% | 93.39459% | 91.26067% | 88.47966% | 85.53476% | 81.88831% | 78.20695% | 73.84110% | 69.60899% | 64.77207% | 60.24501% | 55.23423% | 50.68606% | 45.79144% | 41.46731% | 36.92812% |
| 24 | 100.00000% | 99.99989% | 99.99922% | 99.99630% | 99.98819% | 99.96619% | 99.92247% | 99.82987% | 99.67907% | 99.40716% | 99.02097% | 98.40068% | 97.60490% | 96.43464% | 95.04799% | 93.14683% | 91.03418% | 88.29907% | 85.41715% | 81.86079% | 78.27825% | 74.03407% | 69.92046% | 65.21573% | 60.80620% | 55.91569% | 51.46521% | 46.66127% | 42.40241% |
| 25 | 100.00000% | 99.99995% | 99.99954% | 99.99796% | 99.99259% | 99.97992% | 99.94911% | 99.89267% | 99.78023% | 99.60569% | 99.30255% | 98.88483% | 98.22976% | 97.40566% | 96.21239% | 94.81619% | 92.92071% | 90.83084% | 88.14115% | 85.31972% | 81.84863% | 78.35892% | 74.22865% | 70.22595% | 65.64518% | 61.34629% | 56.56970% | 52.21252% | 47.49621% |
| 26 | 100.00000% | 99.99997% | 99.99975% | 99.99875% | 99.99577% | 99.98716% | 99.96900% | 99.92836% | 99.85855% | 99.72614% | 99.52860% | 99.19612% | 98.74962% | 98.06372% | 97.21538% | 96.00357% | 94.60137% | 92.71433% | 90.64832% | 88.00338% | 85.24003% | 81.84980% | 78.44751% | 74.42422% | 70.52564% | 66.06144% | 61.86698% | 57.19849% | 52.93055% |
| 27 | 100.00000% | 99.99998% | 99.99985% | 99.99931% | 99.99736% | 99.99245% | 99.97982% | 99.95538% | 99.90415% | 99.82056% | 99.66837% | 99.44876% | 99.08891% | 98.61626% | 97.90310% | 97.03410% | 95.80760% | 94.40238% | 92.52600% | 90.48457% | 87.88357% | 85.17601% | 81.86259% | 78.54284% | 74.62028% | 70.81971% | 66.46542% | 62.36975% | 57.80403% |
| 28 | 100.00000% | 99.99999% | 99.99992% | 99.99958% | 99.99850% | 99.99520% | 99.98784% | 99.97048% | 99.93911% | 99.87672% | 99.77920% | 99.60764% | 99.36703% | 98.98176% | 98.48544% | 97.74825% | 96.86176% | 95.62387% | 94.21811% | 92.35418% | 90.33778% | 87.77982% | 85.12588% | 81.88554% | 78.64390% | 74.81643% | 71.10835% | 66.85793% | 62.85591% |
| 29 | 100.00000% | 100.00000% | 99.99995% | 99.99977% | 99.99907% | 99.99720% | 99.99214% | 99.98182% | 99.95912% | 99.92030% | 99.84639% | 99.73497% | 99.54461% | 99.28413% | 98.87537% | 98.35766% | 97.59938% | 96.69819% | 95.45175% | 94.04754% | 92.19746% | 90.20637% | 87.69048% | 85.08810% | 81.91741% | 78.74983% | 75.01232% | 71.39173% | 67.23969% |
And for all you coders, here is the source: