3190:
3210:
36:
715: = 1.833...×1.833... = 3.3611..., or fractional odds of 2.3611 − 1. This represents an implied probability of 29.752% (1/3.3611) and multiplying by 4 (for each of the four equally likely combinations of outcomes) gives a total book of 119.01%. Thus the overround has slightly more than doubled by combining two single bets into a double.
3220:
3200:
956:", which must normally be requested as such on the betting slip) in which the returns from one selection in the accumulator are split to form an equal-stake each-way bet on the next selection and so on until all selections have been used. The first example below shows the two different approaches to settling these types of bets.
514:' or 'vig' and represents the bookmaker's expected profit. Thus, in an "ideal" situation, if the bookmaker accepts £120 in bets at his own quoted odds in the correct proportion, he will pay out only £100 (including returned stakes) no matter the actual outcome of the football match. Examining how he potentially achieves this:
623:
book in each of their matches, i.e. each has a book of 100%, all players would be offered at odds of Evens (1-1). However, a bookmaker would probably offer odds of 5-6 (for example) on each of the two possible outcomes in each event (each tennis match). This results in a book for each of the tennis
168:
The odds quoted for a particular event may be fixed but are more likely to fluctuate in order to take account of the size of wagers placed by the bettors in the run-up to the actual event (e.g. a horse race). This article explains the mathematics of making a book in the (simpler) case of the former
880:
by bookmakers to the nearest penny below. Calculations below for multiple-bet wagers result in totals being shown for the separate categories (e.g. doubles, trebles etc.), and therefore overall returns may not be exactly the same as the amount received from using the computer software available to
770:
This massive increase in potential profit for the bookmaker (19% instead of 9% on an event; in this case the double) is the main reason why bookmakers pay bonuses for the successful selection of winners in multiple bets. Compare offering a 25% bonus on the correct choice of four winners from four
871:
below for detailed example). All bets are taken as 'win' bets unless 'each-way' is specifically stated. All show use of fractional odds: replace (fractional odds + 1) by decimal odds if decimal odds are known. Non-runners are treated as winners with fractional odds of zero (decimal odds of 1).
208:
units they bet. Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. For example, a bet of £40 at 6 − 4 (fractional odds) will pay out £40 + £60 = £100. The equivalent decimal odds are 2.5; £40 × 2.5
447:
book). The bookmaker will reduce these odds to ensure a profit. Consider the simplest model of reducing, which uses a proportional decreasing of odds. For the above example, the following odds are in the same proportion with regard to their implied probabilities (3:2:1):
1642:
For the family of full cover bets that do not include singles an adjustment to the calculation is made to leave just the doubles, trebles and accumulators. Thus, a previously described winning £10 Yankee with winners at 1-3, 5-2, 6-4 and Evens has returns calculated by:
1428:'odds multiplier' is a combined decimal odds value and is the result of all the individual bets that make up a full cover bet, including singles if needed. E.g. if a successful £10 Yankee returned £461.35 then the overall 'odds multiplier' (
948:" method, meaning that the bet consists of a win accumulator and a separate place accumulator (Note: a double or treble is an accumulator with 2 or 3 selections respectively). However, a more uncommon way of settling these type of bets is "
1123:
Note: 'All up to win' means there are insufficient participants in the event for place odds to be given (e.g. 4 or fewer runners in a horse race). The only 'place' therefore is first place, for which the win odds are given.
1651:
In effect, the bet has been calculated as a Lucky 15 minus the singles. Note that the total returns value of £999.16 is a penny higher than the previously calculated value as this quicker method only involves rounding the
673:
or 0.25, and the fractional odds against each one occurring is 3-1. A bet of 100 units on any of the four combinations would produce a return of 100 × (3/1 + 1) = 400 units if successful, reflecting decimal odds of 4.0.
1798:
Note: Expert bet settlers before the introduction of bet-settling software would have invariably used an algebraic-type method together with a simple calculator to determine the return on a bet (see below).
165:
of the various outcomes of an event in a downward fashion (i.e. the bookmaker will pay out using his actual odds, an amount which is less than the true odds would have paid, thus ensuring a profit).
1795:
bets may be calculated in a similar manner as above using the appropriate full cover bet (if sufficient winners) together with the required number of 2 winner- and 1 winner Up and Down bets.
1768:
Note: This calculation of a bet where the stake is not returned is called "receiving the odds to the stake" on the winner; in this case receiving the odds to £10 (on the 5-1 winner).
549:
Bookmaker margin in
English football leagues decreased in recent years. A study of six large bookmakers between the 2005/06 season and 2017/2018 season showed that average margin in
3298:
775:, for example, when the potential overround on a simple fourfold of races with individual books of 120% is over 107% (a book of 207%). This is why bookmakers offer bets such as
847:× 100 - 100. E.g. the previously mentioned fourfold consisting of individual books of 120% (1.20) gives an overround of 1.20 × 1.20 × 1.20 × 1.20 × 100 − 100 = 107.36%.
358:
In considering a football match (the event) that can be either a 'home win', 'draw' or 'away win' (the outcomes) then the following odds might be encountered to represent the
593:
bet, then the effect of the overround in the book of each selection is compounded. This is to the detriment of the punter in terms of the financial return compared to the
1843:+ 1), ... etc. together in the required manner and adding or subtracting additional components. If required, (decimal odds + 1) may be replaced by (fractional odds + 2).
1377:
bet has been shown above. The other named bets are calculated in a similar way by looking at all the possible combinations of selections in their multiples and singles.
1257:
bets have been shown above. The other named bets are calculated in a similar way by looking at all the possible combinations of selections in their multiples. Note: A
1614:
Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners and a further placed horse can be relatively easily calculated as follows:
1384:
win, then the remaining winners are treated as being a wholly successful bet on the next 'family member' down. For example, only two winners out of three in a
1268:
win, then the remaining winners are treated as being a wholly successful bet on the next 'family member' down. For example, only two winners out of three in a
855:
In settling winning bets, either decimal odds are used, or one is added to the fractional odds. This is to include the stake in the return. The place part of
663:. As each of the outcomes for this example have been deliberately chosen to ensure that they are equally likely, the probability of each outcome occurring is
2342:
1051:" provides greater compensation if one selection is a loser as each of the other winners provide a greater amount of place money for subsequent selections.
158:
153:
A bookmaker strives to accept bets on the outcome of an event in the right proportions in order to make a profit regardless of which outcome prevails. See
1709:
In the days before software became available for use by bookmakers and those settling bets in
Licensed Betting Offices (LBOs) this method was virtually
46:
3314:
3279:
2641:
2347:
1361:
1355:
1349:
1315:
1241:
1235:
1229:
1223:
1185:
1154:
1077:
1060:
965:
895:
864:
860:
856:
789:
783:
777:
772:
590:
586:
582:
677:
The decimal odds of a multiple bet is often calculated by multiplying the decimal odds of the individual bets, the idea being that if the events are
3256:
2659:
3289:
1027:
Returns (first selection) = £100 × (2/1 + 1) + £100 × (2/5 + 1) = £440 which is split equally to give a £220 each-way bet on the second selection)
1782:
A Round Robin with 2 winners is calculated as a double plus one Up and Down bet with 2 winners plus two Up and Down bets with 1 winner in each.
3062:
2566:
2547:
3067:
1471:+ 1)... etc. together in the required manner and subtracting 1. If required, (decimal odds + 1) may be replaced by (fractional odds + 2).
2575:
104:
3072:
2703:
566:
76:
137:
on the various possible outcomes of a single event. The phrase originates from the practice of recording such wagers in a hard-bound
3294:
2675:
682:
678:
3223:
1300:. Virtually all bookmakers use computer software for ease, speed and accuracy of calculation for the settling of multiples bets.
528:
Total stakes received are £120.00 with a maximum payout of £100.00 irrespective of the result. This £20.00 profit represents a 16
83:
3028:
61:
2668:
Probability Guide to
Gambling: The Mathematics of dice, slots, roulette, baccarat, blackjack, poker, lottery and sport bets
90:
3335:
3049:
2721:
3304:
3274:
3249:
3077:
2613:
2620:"Why is There a Favourite-Longshot Bias in British Racetrack Betting Markets", Leighton Vaughan Williams, David Paton.
2940:
2845:
1713:
for saving time and avoiding the multiple repetitious calculations necessary in settling bets of the full cover type.
72:
611:, both players are assessed to have an equal chance of winning. The situation is the same in Match 2 between players
647:
There are four possible outcomes from combining the results from both matches: the winning pair of players could be
541:
In reality, bookmakers use models of reducing that are more complicated than this model of the 'ideal' situation.
1373:
which have all possible multiples present together with single bets on all selections. An examples of a winning
176:
It is important to understand the relationship between fractional and decimal odds. Fractional odds are written
2945:
2882:
252:
It is also important to understand the relationship between odds and implied probabilities: Fractional odds of
685:
of the individual bets. In the above case with fractional odds of 5 − 6, the decimal odds are
3242:
3126:
3100:
2633:
1779:
A Round Robin with 3 winners is calculated as a Trixie plus three Up and Down bets with 2 winners in each.
794:
offering double the odds for one winner and increasing percentage bonuses for two, three and more winners.
510:
The amount by which the actual 'book' exceeds 100% is known as the 'overround', 'bookmaker margin' or the '
2898:
2696:
2473:"Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds"
379:
These odds can be represented as implied probabilities (or percentages by multiplying by 100) as follows:
3309:
3095:
2957:
2913:
2650:
1133:
Returns (place fivefold) = £100 × (1/4 + 1) × (11/40 + 1) × (5/16 + 1) × (1/2 + 1) × (3/5 + 1) = £502.03
1404:. The place part of each-way bets is calculated separately using reduced place odds. Thus, an each-way
1288:. The place part of each-way bets is calculated separately using reduced place odds. Thus, an each-way
316:= 0.4 (40%). An implied probability of x is represented by fractional odds of (1 −
97:
3209:
3203:
3175:
2967:
2935:
2903:
2840:
2785:
2622:
2593:
1606:
E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following:
1128:
Returns (win fivefold) = £100 × (1/1 + 1) × (11/8 + 1) × (5/4 + 1) × (1/2 + 1) × (3/1 + 1) = £6412.50
3023:
2993:
2977:
2877:
2648:"Information, Prices and Efficiency in a Fixed-Odds Betting Market", Peter F. Pope, David A. Peel,
2632:, by John Fingleton and Patrick Waldron, Trinity Economic Paper Series, Technical Paper No. 96/9,
1751:
the odds; i.e. £20 singles at 7-1 and 15-4 and is the preferred manual way of calculating the bet.
3114:
3057:
3044:
3018:
3013:
3008:
2998:
2972:
2962:
2952:
2872:
2835:
2813:
170:
2521:
2611:"Tests of the Efficiency of Racetrack Betting Using Bookmaker Odds", Ron Bird, Michael McCrae,
1424:
Returns on any bet may be considered to be calculated as 'stake unit' × 'odds multiplier'. The
3340:
3234:
3193:
3003:
2930:
2925:
2918:
2908:
2795:
2689:
2671:
2543:
2448:
324:, e.g. 0.2 is (1 − 0.2)/0.2 = 0.8/0.2 = 4/1 (4-1, 4 to 1) (equivalently,
3170:
2805:
2762:
2571:
1567:. Therefore to calculate the returns for a winning Patent it is just a case of multiplying (
554:
134:
53:
1785:
A Round Robin with 1 winner is calculated as two Up and Down bets with one winner in each.
1737:
Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 × 15/8 + £20 × (7/2 + 1) = £127.50
1732:
Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 × 7/2 + £20 × (15/8 + 1) = £127.50
2740:
2585:
1388:
means the bet is settled as a double and two singles; only three winners out of four in a
3139:
2830:
2422:
The Art of
Legging: The History, Theory, and Practice of Bookmaking on the English Turf
1370:
1292:
on seven horses with three winners and a further two placed horses is settled as a win
1246:
562:
558:
550:
2472:
600:
For example, consider a double made by selecting the winners from two tennis matches:
443:
By adding the percentages together a total 'book' of 100% is achieved (representing a
3329:
3144:
2818:
2747:
2495:
1408:
on six horses with three winners and a further two placed horses is settled as a win
1203:
Returns (1 fourfold) = £10 × (1/3 + 1) × (5/2 + 1) × (6/4 + 1) × (1/1 + 1) = £233.33
3149:
2825:
3134:
767:= 1.0909 × 1.0909 × 100 - 100 = 19.01%.
35:
3165:
2601:
367:
154:
2452:
161:. This is achieved primarily by adjusting what are determined to be the true
3213:
2532:
142:
2600:"Carnap and de Finetti on Bets and the Probability of Singular Events: The
1032:
Returns (second selection) = £220 × (5/4 + 1) + £220 × (5/16 + 1) = £783.75
2752:
2712:
1758:
E.g. £10 Up and Down with a winner at 5-1 and a loser; total staked = £20
1272:
means the bet is settled as a double; only four winners out of five in a
597:
of all of the selections winning and thus resulting in a successful bet.
511:
126:
17:
2436:
1763:
Returns (£10 single at 5-1 ATC £10 single on 'loser') = £10 × 5/1 = £50
1167:
Returns (1 treble) = £10 × (4/7 + 1) × (2/1 + 1) × (11/10 + 1) = £99.00
1047:" will always provide a greater return if all selections win, whereas "
863:
part; the method is identical but the odds are reduced by whatever the
725:), expressed as a percentage, is calculated from the individual books
209:= £100. We can convert fractional to decimal odds by the formula
2757:
2735:
1727:
E.g. £20 Up and Down with winners at 7-2 and 15-8; total staked = £40
1603:. Now multiply by the unit stake to get the total return on the bet.
1333:
Returns (1 treble) = £2 × (4/6 + 1) × (2/1 + 1) × (11/4 + 1) = £37.50
138:
2630:
Optimal
Determination of Bookmakers' Betting Odds: Theory and Tests
521:
A stake of £40.00 @ 6-4 returns £100.00 (exactly) for a drawn match
1827:, i.e. (fractional odds + 1), then an 'odds multiplier'
1632:
or more simply as 0.50 × (2.4 × 2.5 × 2.7 × 2.375 × 2.9 − 1)
524:
A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win
518:
A stake of £60.00 @ 4-6 returns £100.00 (exactly) for a home win.
1831:
can be calculated algebraically by multiplying the expressions (
1459:
can be calculated algebraically by multiplying the expressions (
1261:
may be thought of as a full cover bet with only two selections.
1006:
Returns (place double) = £100 × (2/5 + 1) × (5/16 + 1) = £183.75
162:
3238:
2685:
1249:
which have all possible multiples present. Examples of winning
200:), meaning a winning bettor will receive their money back plus
2626:, volume 107, number 440 (January, 1997), pages 150–158.
29:
2654:, volume 56, number 223, (August, 1989), pages 323–341.
2617:, volume 33, number 12 (December, 1987), pages 152–156.
2608:, volume 29, number 4. (December, 1978), pages 325–346.
935:
Total returns if selection wins = £530; if only placed = £155
1188:
with winners at 1-3, 5-2, 6-4 and Evens; total staked = £110
944:
Each-way multiple bets are usually settled using a default "
2681:
1067:
Returns = £100 × (3/1 + 1) × (4/6 + 1) × (11/4 + 1) = £2500
145:
for the person laying the bets and thus 'making the book'.
2542:", Ferde Rombola, California: Romford Press, 1984, 147pp.
2500:
Mathematical
Methods in Economics - Conference Proceedings
1001:
Returns (win double) = £100 × (2/1 + 1) × (5/4 + 1) = £675
801:
selections, the combined percentage overround of books of
241: = 1) can be obtained from decimal odds by
2657:"A Mathematical Perspective on Gambling", Molly Maxwell,
930:
Returns (place) = £100 × (11/20 + 1) = £100 × 1.55 = £155
760: × 100 - 100. In the example we have
2597:, volume 51, number 3. (May, 1978), pages 188–192.
1622:
or more simply as 0.50 × (4 × 4.5 × 5.5 × 3.5 − 1)
383:
Evens (or 1-1) corresponds to an implied probability of
1063:
with winners at 3-1, 4-6 and 11-4; total staked = £100
57:
1157:
with winners at 4-7, 2-1 and 11-10; total staked = £40
925:
Returns (win) = £100 × (11/4 + 1) = £100 × 3.75 = £375
911:
E.g. £100 each-way single at 11 − 4 (
581:) combines more than one selection in, for example, a
2524:, US-SC-North Charleston: Create Space., 2011, 144pp.
1318:
with winners at 4-6, 2-1 and 11-4; total staked = £14
141:(the 'book') and gives the English language the term
2496:"Bookmakers' Efficiency in English Football Leagues"
3158:
3125:
3109:
3088:
3037:
2986:
2891:
2865:
2858:
2804:
2778:
2771:
2728:
507:percentages together a 'book' of 120% is achieved.
1656:answer, and not rounding at each individual step.
2645:, 1994, volume 13, number 4, pages 145–147.
2606:The British Journal for the Philosophy of Science
1747:Note: This is the same as two £20 single bets at
148:
2535:, London: Morrison & Gibb Ltd., 1948, 160pp.
797:In general, for any accumulator bet from two to
718:In general, the combined overround on a double (
2415:
2413:
2411:
2409:
2407:
2405:
2403:
2401:
2399:
2397:
2381:
2379:
2377:
2375:
2373:
2371:
2369:
2367:
2365:
2363:
902:Returns = £100 × (9/2 + 1) = £100 × 5.5 = £550
898:at 9 − 2; total staked = £100
3250:
2697:
2556:, Malcolm Boyle, High Stakes Publishing 2006.
824:given in terms of decimals, is calculated by
489:4-1 corresponds to an implied probability of
476:6-4 corresponds to an implied probability of
463:4-6 corresponds to an implied probability of
419:5-1 corresponds to an implied probability of
396:2-1 corresponds to an implied probability of
149:Making a 'book' (and the notion of overround)
8:
2670:", Catalin Barboianu, Infarom, 2006, 316pp.
2343:Statistical association football predictions
624:matches of 109.09...%, calculated by 100 × (
545:Bookmaker margin in English football leagues
62:introducing citations to additional sources
1579:+ 1) together and subtracting 1 to get the
695:. So the decimal odds of the double bet is
681:then the implied probability should be the
538: % profit on turnover (20.00/120.00).
159:coherence (philosophical gambling strategy)
3257:
3243:
3235:
2862:
2775:
2704:
2690:
2682:
2639:"Odds That Don't Add Up!", Mike Fletcher,
2437:"Victorian Bookmakers and Their Customers"
334: − 1 to 1), and decimal odds of
2642:Teaching Mathematics and its Applications
2591:"Maximin Hedges", Jean-Claude Derderian,
2348:Glossary of bets offered by UK bookmakers
2663:, volume 1, (1999), pages 123–132.
2660:MIT Undergraduate Journal of Mathematics
2466:
2464:
2462:
1380:Should a selection in one of these bets
1264:Should a selection in one of these bets
881:bookmakers to calculate total winnings.
867:factor is for the particular event (see
52:Relevant discussion may be found on the
2359:
1628:Returns (place part) = 0.50 × = £11.79
859:bets is calculated separately from the
1618:Returns (win part) = 0.50 × = £172.75
1455:, i.e. (fractional odds + 1), then an
1280:; only five winners out of eight in a
1198:Returns (4 trebles) = £10 × = £451.66
1193:Returns (6 doubles) = £10 × = £314.16
1162:Returns (3 doubles) = £10 × = £143.14
362:chance of each of the three outcomes:
264:) represent an implied probability of
2634:Trinity College, University of Dublin
2604:Argument Reconsidered" Klaus Heilig,
2567:The Mathematics of Games and Gambling
1110:odds), 1-2 (all up to win) and 3-1 (
7:
3199:
1717:Settling other types of winning bets
1396:; only four winners out of six in a
1369:bets form a family of bets known as
1328:Returns (3 doubles) = £2 × = £45.00
1323:Returns (3 singles) = £2 × = £16.83
683:product of the implied probabilities
555:English Football League Championship
3219:
2576:Mathematical Association of America
988:odds a place); total staked = £200
921:odds a place); total staked = £200
169:event. For the second method, see
25:
2494:Marek, Patrice (September 2018).
2424:(3rd ed.). Rotex Publishing.
260:(with corresponding decimal odds
3218:
3208:
3198:
3189:
3188:
2562:, Michael Adams, Raceform, 2002.
1663:for the Yankee bet is given by:
876:winnings are invariably rounded
45:relies largely or entirely on a
34:
2080:4 selections with decimal odds
1906:3 selections with decimal odds
1852:2 selections with decimal odds
1531:+ 1. This is equivalent to the
1479:3 selections with decimal odds
1245:form a family of bets known as
1610:Total returns = £2 × = £99.33
1:
2560:Secrets of Successful Betting
2477:Journal of Prediction Markets
2435:Itzkowitz, David C. (1988).
1371:full cover bets with singles
1304:Full cover bets with singles
1076:E.g. £100 each-way fivefold
739:, expressed as decimals, by
553:decreased from 9% to 4%, in
233:. Hence, fractional odds of
2582:The Mathematics of Gambling
1120:odds); total staked = £200
603:In Match 1 between players
563:English Football League Two
559:English Football League One
73:"Mathematics of bookmaking"
3357:
3315:three-phase electric power
3280:artificial neural networks
1583:for the winning bet, i.e.
1045:Win to Win, Place to Place
994:Win to Win, Place to Place
946:Win to Win, Place to Place
573:Overround on multiple bets
296:, e.g. 6-4 corresponds to
3270:
3265:"Mathematics of" articles
3184:
2791:Mathematics of bookmaking
2719:
1503:+ 1) algebraically gives
1400:means it is settled as a
1392:means it is settled as a
1284:means it is settled as a
1276:means it is settled as a
2471:Cortis, Dominic (2015).
2420:Sidney, Charles (2003).
2390:. Maxline International.
2386:Sidney, Charles (1976).
2256:+ 1) − 1 − (
2191:+ 1) − 1 − (
2120:+ 1) − 1 − (
1991:+ 1) − 1 − (
1938:+ 1) − 1 − (
1803:Algebraic interpretation
1688:+ 1) − 1 − (
1420:Algebraic interpretation
1138:Total returns = £6914.53
1080:with winners at Evens (
978:odds a place) and 5-4 (
644:) i.e. 9.09% overround.
3290:cyclic redundacy checks
1742:Total returns = £255.00
1659:In algebraic terms the
1638:Total returns = £184.54
1208:Total returns = £999.15
1172:Total returns = £242.14
1037:Total returns = £783.85
1011:Total returns = £858.75
565:from 11% to 6%, and in
2540:The book on bookmaking
1860:in an Up and Down bet.
1535:for a Patent (treble:
1338:Total returns = £99.33
1090:odds a place), 11-8 (
968:with winners at 2-1 (
872:Fractions of pence in
2651:Economica, New Series
2554:The Art of Bookmaking
1049:Each-Way all Each-Way
1020:Each-Way all Each-Way
950:Each-Way all Each-Way
851:Settling winning bets
27:Theory of laying bets
3336:Gambling mathematics
3176:Video game addiction
2786:Gambling mathematics
2722:Gambling terminology
2623:The Economic Journal
2594:Mathematics Magazine
2272:− 1) + 2 × =
237: − 1 (ie.
58:improve this article
964:E.g. £100 each-way
133:is the practice of
3295:general relativity
2836:Due Column betting
2814:Advantage gambling
2614:Management Science
2588:, L. Stuart, 1984.
2529:How to make a book
2388:The Art of Legging
2319:(1 winner) = 3 × (
2294:or more simply as
2062:(1 winner) = 2 × (
2037:or more simply as
1823:... represent the
1451:... represent the
171:Parimutuel betting
3323:
3322:
3232:
3231:
3121:
3120:
2854:
2853:
2841:Labouchère system
2796:Poker probability
2548:978-0-935536-37-9
2441:Victorian Studies
2313:
2056:
1918:in a Round Robin.
1647:£10 × ] = £999.16
1633:
1623:
123:
122:
108:
16:(Redirected from
3348:
3259:
3252:
3245:
3236:
3222:
3221:
3212:
3202:
3201:
3192:
3191:
3171:Problem gambling
2863:
2776:
2763:Riverboat casino
2706:
2699:
2692:
2683:
2572:Edward W. Packel
2504:
2503:
2491:
2485:
2484:
2468:
2457:
2456:
2432:
2426:
2425:
2417:
2392:
2391:
2383:
2293:
2036:
1868:(2 winners) = (2
1631:
1621:
1119:
1118:
1114:
1109:
1108:
1104:
1099:
1098:
1094:
1089:
1088:
1084:
987:
986:
982:
977:
976:
972:
920:
919:
915:
771:selections in a
714:
713:
709:
704:
703:
699:
694:
693:
689:
672:
671:
667:
643:
642:
638:
633:
632:
628:
569:from 11% to 8%.
537:
536:
532:
498:
497:
493:
485:
484:
480:
472:
471:
467:
438:
437:
433:
428:
427:
423:
415:
414:
410:
405:
404:
400:
392:
391:
387:
349:
348:
342:
333:
332:
328:
315:
314:
310:
305:
304:
300:
295:
294:
288:
283:
282:
270:
249: − 1.
232:
231:
225:
204:units for every
118:
115:
109:
107:
66:
38:
30:
21:
3356:
3355:
3351:
3350:
3349:
3347:
3346:
3345:
3326:
3325:
3324:
3319:
3266:
3263:
3233:
3228:
3180:
3154:
3117:
3105:
3084:
3078:Atlantic Canada
3033:
2982:
2887:
2850:
2800:
2767:
2724:
2715:
2710:
2586:Edward O. Thorp
2518:Finding an Edge
2513:
2511:Further reading
2508:
2507:
2493:
2492:
2488:
2470:
2469:
2460:
2434:
2433:
2429:
2419:
2418:
2395:
2385:
2384:
2361:
2356:
2339:
2334:
2333:
2292:
2248:(2 winners) = (
2179:(3 winners) = (
2137:
2104:(4 winners) = (
2097:
2077:
2076:
2035:
1983:(2 winners) = (
1926:(3 winners) = (
1919:
1903:
1902:
1872:− 1) + (2
1861:
1849:
1805:
1772:
1771:
1759:
1755:
1754:
1728:
1719:
1477:
1422:
1342:
1341:
1319:
1306:
1247:full cover bets
1212:
1211:
1189:
1176:
1175:
1158:
1145:
1143:Full-cover bets
1116:
1112:
1111:
1106:
1102:
1101:
1096:
1092:
1091:
1086:
1082:
1081:
1041:
1040:
1023:
1015:
1014:
997:
984:
980:
979:
974:
970:
969:
954:Equally Divided
942:
917:
913:
912:
907:Each-way single
887:
853:
846:
837:
830:
823:
814:
807:
766:
759:
752:
745:
738:
731:
724:
711:
707:
706:
701:
697:
696:
691:
687:
686:
669:
665:
664:
640:
636:
635:
630:
626:
625:
577:When a punter (
575:
567:National League
547:
534:
530:
529:
495:
491:
490:
482:
478:
477:
469:
465:
464:
435:
431:
430:
425:
421:
420:
412:
408:
407:
402:
398:
397:
389:
385:
384:
356:
344:
340:
339:
330:
326:
325:
312:
308:
307:
302:
298:
297:
290:
286:
285:
272:
266:
265:
227:
215:
214:
151:
119:
113:
110:
67:
65:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3354:
3352:
3344:
3343:
3338:
3328:
3327:
3321:
3320:
3318:
3317:
3312:
3307:
3302:
3292:
3287:
3282:
3277:
3271:
3268:
3267:
3264:
3262:
3261:
3254:
3247:
3239:
3230:
3229:
3227:
3226:
3216:
3206:
3196:
3185:
3182:
3181:
3179:
3178:
3173:
3168:
3162:
3160:
3156:
3155:
3153:
3152:
3147:
3142:
3140:Game of chance
3137:
3131:
3129:
3123:
3122:
3119:
3118:
3113:
3111:
3107:
3106:
3104:
3103:
3098:
3092:
3090:
3086:
3085:
3083:
3082:
3081:
3080:
3075:
3073:Western Canada
3070:
3065:
3060:
3052:
3047:
3041:
3039:
3035:
3034:
3032:
3031:
3029:United Kingdom
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2990:
2988:
2984:
2983:
2981:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2949:
2948:
2938:
2933:
2928:
2923:
2922:
2921:
2916:
2906:
2901:
2895:
2893:
2889:
2888:
2886:
2885:
2880:
2875:
2869:
2867:
2860:
2856:
2855:
2852:
2851:
2849:
2848:
2843:
2838:
2833:
2831:Asian handicap
2828:
2823:
2822:
2821:
2810:
2808:
2802:
2801:
2799:
2798:
2793:
2788:
2782:
2780:
2773:
2769:
2768:
2766:
2765:
2760:
2755:
2750:
2745:
2744:
2743:
2732:
2730:
2726:
2725:
2720:
2717:
2716:
2711:
2709:
2708:
2701:
2694:
2686:
2680:
2679:
2664:
2655:
2646:
2637:
2627:
2618:
2609:
2598:
2589:
2579:
2563:
2557:
2551:
2536:
2525:
2512:
2509:
2506:
2505:
2486:
2458:
2427:
2393:
2358:
2357:
2355:
2352:
2351:
2350:
2345:
2338:
2335:
2332:
2331:
2314:
2243:
2211:− 1) + (
2207:− 1) + (
2174:
2098:
2079:
2078:
2075:
2074:
2057:
2011:− 1) + (
2007:− 1) + (
1978:
1920:
1905:
1904:
1901:
1900:
1888:
1862:
1851:
1850:
1848:
1845:
1804:
1801:
1770:
1769:
1765:
1764:
1760:
1757:
1756:
1753:
1752:
1744:
1743:
1739:
1738:
1734:
1733:
1729:
1726:
1725:
1718:
1715:
1707:
1706:
1649:
1648:
1640:
1639:
1635:
1634:
1629:
1625:
1624:
1619:
1612:
1611:
1601:+ 1) − 1
1476:
1473:
1421:
1418:
1340:
1339:
1335:
1334:
1330:
1329:
1325:
1324:
1320:
1313:
1312:
1305:
1302:
1210:
1209:
1205:
1204:
1200:
1199:
1195:
1194:
1190:
1183:
1182:
1174:
1173:
1169:
1168:
1164:
1163:
1159:
1152:
1151:
1144:
1141:
1140:
1139:
1135:
1134:
1130:
1129:
1069:
1068:
1039:
1038:
1034:
1033:
1029:
1028:
1024:
1017:
1016:
1013:
1012:
1008:
1007:
1003:
1002:
998:
991:
990:
941:
938:
937:
936:
932:
931:
927:
926:
904:
903:
886:
883:
852:
849:
842:
835:
828:
819:
812:
805:
764:
757:
750:
743:
736:
729:
722:
574:
571:
551:Premier League
546:
543:
526:
525:
522:
519:
501:
500:
487:
474:
460:
459:
456:
453:
441:
440:
417:
394:
377:
376:
373:
370:
355:
352:
150:
147:
121:
120:
56:. Please help
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3353:
3342:
3339:
3337:
3334:
3333:
3331:
3316:
3313:
3311:
3308:
3306:
3305:paper folding
3303:
3300:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3275:apportionment
3273:
3272:
3269:
3260:
3255:
3253:
3248:
3246:
3241:
3240:
3237:
3225:
3217:
3215:
3211:
3207:
3205:
3197:
3195:
3187:
3186:
3183:
3177:
3174:
3172:
3169:
3167:
3164:
3163:
3161:
3157:
3151:
3148:
3146:
3145:Game of skill
3143:
3141:
3138:
3136:
3133:
3132:
3130:
3128:
3124:
3116:
3112:
3110:South America
3108:
3102:
3099:
3097:
3094:
3093:
3091:
3087:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3055:
3053:
3051:
3050:United States
3048:
3046:
3043:
3042:
3040:
3038:North America
3036:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2991:
2989:
2985:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2947:
2944:
2943:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2920:
2917:
2915:
2912:
2911:
2910:
2907:
2905:
2902:
2900:
2897:
2896:
2894:
2890:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2868:
2864:
2861:
2857:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2820:
2819:Card counting
2817:
2816:
2815:
2812:
2811:
2809:
2807:
2803:
2797:
2794:
2792:
2789:
2787:
2784:
2783:
2781:
2777:
2774:
2770:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2748:Online casino
2746:
2742:
2739:
2738:
2737:
2734:
2733:
2731:
2727:
2723:
2718:
2714:
2707:
2702:
2700:
2695:
2693:
2688:
2687:
2684:
2677:
2676:973-87520-3-5
2673:
2669:
2665:
2662:
2661:
2656:
2653:
2652:
2647:
2644:
2643:
2638:
2635:
2631:
2628:
2625:
2624:
2619:
2616:
2615:
2610:
2607:
2603:
2599:
2596:
2595:
2590:
2587:
2583:
2580:
2577:
2573:
2569:
2568:
2564:
2561:
2558:
2555:
2552:
2549:
2545:
2541:
2537:
2534:
2530:
2526:
2523:
2519:
2515:
2514:
2510:
2501:
2497:
2490:
2487:
2482:
2478:
2474:
2467:
2465:
2463:
2459:
2454:
2450:
2446:
2442:
2438:
2431:
2428:
2423:
2416:
2414:
2412:
2410:
2408:
2406:
2404:
2402:
2400:
2398:
2394:
2389:
2382:
2380:
2378:
2376:
2374:
2372:
2370:
2368:
2366:
2364:
2360:
2353:
2349:
2346:
2344:
2341:
2340:
2336:
2330:
2328:
2323:− 1) =
2322:
2318:
2315:
2312:
2310:
2306:
2302:
2297:
2291:
2289:
2285:
2281:
2277:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2244:
2242:
2240:
2236:
2232:
2228:
2224:
2220:
2215:− 1) =
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2175:
2173:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2100:
2099:
2095:
2091:
2087:
2083:
2073:
2071:
2066:− 1) =
2065:
2061:
2058:
2055:
2053:
2049:
2045:
2040:
2034:
2032:
2028:
2024:
2020:
2015:− 1) =
2014:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1979:
1977:
1975:
1971:
1967:
1963:
1959:
1955:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1922:
1921:
1917:
1913:
1909:
1899:
1897:
1893:(1 winner) =
1892:
1889:
1887:
1885:
1881:
1876:− 1) =
1875:
1871:
1867:
1864:
1863:
1859:
1855:
1846:
1844:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1802:
1800:
1796:
1794:
1790:
1786:
1783:
1780:
1777:
1776:
1767:
1766:
1762:
1761:
1750:
1746:
1745:
1741:
1740:
1736:
1735:
1731:
1730:
1724:
1723:
1716:
1714:
1712:
1705:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1675:
1669:
1666:
1665:
1664:
1662:
1657:
1655:
1646:
1645:
1644:
1637:
1636:
1630:
1627:
1626:
1620:
1617:
1616:
1615:
1609:
1608:
1607:
1604:
1602:
1600:
1596:
1592:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1491:. Expanding (
1490:
1486:
1482:
1474:
1472:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1433:
1432:) is 46.135.
1431:
1427:
1419:
1417:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1378:
1376:
1372:
1368:
1364:
1363:
1358:
1357:
1352:
1351:
1346:
1337:
1336:
1332:
1331:
1327:
1326:
1322:
1321:
1317:
1311:
1310:
1303:
1301:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1262:
1260:
1256:
1252:
1248:
1244:
1243:
1238:
1237:
1232:
1231:
1226:
1225:
1220:
1216:
1207:
1206:
1202:
1201:
1197:
1196:
1192:
1191:
1187:
1181:
1180:
1171:
1170:
1166:
1165:
1161:
1160:
1156:
1150:
1149:
1142:
1137:
1136:
1132:
1131:
1127:
1126:
1125:
1121:
1100:odds), 5-4 (
1079:
1074:
1073:
1066:
1065:
1064:
1062:
1057:
1056:
1052:
1050:
1046:
1036:
1035:
1031:
1030:
1026:
1025:
1021:
1010:
1009:
1005:
1004:
1000:
999:
995:
989:
967:
962:
961:
957:
955:
952:" (known as "
951:
947:
940:Multiple bets
939:
934:
933:
929:
928:
924:
923:
922:
909:
908:
901:
900:
899:
897:
892:
891:
884:
882:
879:
875:
870:
866:
862:
858:
850:
848:
845:
841:
834:
827:
822:
818:
811:
804:
800:
795:
793:
791:
786:
785:
780:
779:
774:
768:
763:
756:
753: ×
749:
742:
735:
728:
721:
716:
684:
680:
675:
662:
658:
654:
650:
645:
622:
618:
614:
610:
606:
601:
598:
596:
592:
588:
584:
580:
572:
570:
568:
564:
560:
556:
552:
544:
542:
539:
523:
520:
517:
516:
515:
513:
508:
506:
488:
475:
462:
461:
457:
454:
451:
450:
449:
446:
418:
395:
382:
381:
380:
374:
371:
369:
365:
364:
363:
361:
353:
351:
347:
338: =
337:
323:
319:
293:
280:
276:
269:
263:
259:
256: −
255:
250:
248:
245: =
244:
240:
236:
230:
223:
219:
213: =
212:
207:
203:
199:
195:
191:
187:
183:
180: −
179:
174:
172:
166:
164:
160:
156:
146:
144:
140:
136:
132:
131:making a book
128:
117:
114:February 2012
106:
103:
99:
96:
92:
89:
85:
82:
78:
75: –
74:
70:
69:Find sources:
63:
59:
55:
49:
48:
47:single source
43:This article
41:
37:
32:
31:
19:
3299:introduction
3284:
3150:List of bets
2883:South Africa
2826:Dice control
2790:
2667:
2658:
2649:
2640:
2629:
2621:
2612:
2605:
2592:
2581:
2565:
2559:
2553:
2539:
2528:
2517:
2499:
2489:
2480:
2476:
2444:
2440:
2430:
2421:
2387:
2326:
2324:
2320:
2316:
2308:
2304:
2300:
2299:
2295:
2287:
2283:
2279:
2275:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:) − 10
2238:
2234:
2230:
2226:
2222:
2218:
2216:
2212:
2208:
2204:
2203:) + 2 × + (
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:) − 13
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2139:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2093:
2089:
2085:
2081:
2069:
2067:
2063:
2059:
2051:
2047:
2043:
2042:
2038:
2030:
2026:
2022:
2018:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1973:
1969:
1965:
1961:
1957:
1953:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1915:
1911:
1907:
1895:
1894:
1890:
1883:
1879:
1877:
1873:
1869:
1865:
1857:
1853:
1840:
1836:
1832:
1828:
1825:decimal odds
1824:
1820:
1816:
1812:
1808:
1806:
1797:
1792:
1788:
1787:
1784:
1781:
1778:
1774:
1773:
1748:
1721:
1720:
1710:
1708:
1701:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1671:
1667:
1660:
1658:
1653:
1650:
1641:
1613:
1605:
1598:
1594:
1590:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1478:
1468:
1464:
1460:
1456:
1453:decimal odds
1452:
1448:
1444:
1440:
1436:
1434:
1429:
1425:
1423:
1413:
1412:and a place
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1379:
1374:
1366:
1360:
1354:
1348:
1344:
1343:
1308:
1307:
1297:
1296:and a place
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1263:
1258:
1254:
1250:
1240:
1234:
1228:
1222:
1218:
1214:
1213:
1178:
1177:
1147:
1146:
1122:
1075:
1071:
1070:
1058:
1054:
1053:
1048:
1044:
1042:
1019:
993:
963:
959:
958:
953:
949:
945:
943:
910:
906:
905:
893:
889:
888:
877:
873:
868:
854:
843:
839:
832:
825:
820:
816:
809:
802:
798:
796:
788:
782:
776:
769:
761:
754:
747:
740:
733:
726:
719:
717:
676:
660:
656:
652:
648:
646:
620:
616:
612:
608:
604:
602:
599:
594:
578:
576:
548:
540:
527:
509:
504:
502:
444:
442:
378:
359:
357:
345:
335:
321:
317:
291:
278:
274:
267:
261:
257:
253:
251:
246:
242:
238:
234:
228:
221:
217:
210:
205:
201:
197:
193:
189:
185:
181:
177:
175:
167:
152:
130:
124:
111:
101:
94:
87:
80:
68:
44:
3224:WikiProject
3135:Casino game
3101:New Zealand
2941:Philippines
2779:Mathematics
2447:(1): 7–30.
2311:) − 6
2290:) − 7
2054:) − 4
2033:) − 5
1976:) − 7
1950:) + 2 × =
1775:Round Robin
1722:Up and down
1551:; singles:
1539:; doubles:
1365:and higher
1290:Super Heinz
1236:Super Heinz
1078:accumulator
1072:Accumulator
869:Accumulator
679:independent
591:accumulator
135:laying bets
3330:Categories
3285:bookmaking
3214:Wiktionary
3166:Gaming law
2899:Bangladesh
2846:Martingale
2806:Strategies
2602:Dutch Book
2522:Ron Loftus
2502:: 330–335.
2354:References
2329:− 1)
2096:in a Flag.
2072:− 1)
1886:− 1)
1793:Super Flag
1711:de rigueur
1575:+ 1) and (
1059:E.g. £100
894:E.g. £100
890:Win single
503:By adding
155:Dutch book
129:parlance,
84:newspapers
3096:Australia
2958:Singapore
2914:Hong Kong
2859:By region
2533:Phil Bull
2453:0042-5222
2282:+ 1) + 3(
2264:) + 2 × (
2229:+ 1) + 4(
2156:+ 1) + 5(
2025:+ 1) + 2(
1999:) + 2 × (
1964:+ 1) + 3(
1898:− 1
1184:E.g. £10
1153:E.g. £10
595:true odds
458:Away: 4-1
455:Draw: 6-4
452:Home: 4-6
375:Away: 5-1
372:Draw: 2-1
143:bookmaker
54:talk page
18:Overround
3341:Wagering
3194:Category
2968:Thailand
2936:Pakistan
2904:Cambodia
2753:Cardroom
2713:Gambling
2337:See also
2252:+ 1) × (
2187:+ 1) × (
2183:+ 1) × (
2136:) + 2 ×
2116:+ 1) × (
2112:+ 1) × (
2108:+ 1) × (
1987:+ 1) × (
1934:+ 1) × (
1930:+ 1) × (
1847:Examples
1475:Examples
1414:Lucky 31
1406:Lucky 63
1402:Lucky 15
1398:Lucky 63
1390:Lucky 15
1362:Lucky 63
1356:Lucky 31
1350:Lucky 15
1314:E.g. £2
1298:Canadian
1286:Canadian
1274:Canadian
1224:Canadian
857:each-way
838:× ... ×
790:Lucky 63
784:Lucky 31
778:Lucky 15
512:vigorish
127:gambling
3204:Commons
3089:Oceania
3063:Ontario
3054:Canada
3024:Ukraine
2994:Estonia
2978:Vietnam
2878:Nigeria
2772:Science
2636:, 1999.
2578:, 2006.
1839:+ 1), (
1835:+ 1), (
1571:+ 1), (
1467:+ 1), (
1463:+ 1), (
1426:overall
1282:Goliath
1242:Goliath
1115:⁄
1105:⁄
1095:⁄
1085:⁄
1043:Note: "
983:⁄
973:⁄
916:⁄
885:Singles
815:, ...,
746:=
710:⁄
700:⁄
690:⁄
668:⁄
639:⁄
629:⁄
619:. In a
533:⁄
494:⁄
481:⁄
468:⁄
434:⁄
424:⁄
411:⁄
401:⁄
388:⁄
354:Example
343:⁄
329:⁄
311:⁄
306:=
303:(6 + 4)
301:⁄
289:⁄
284:=
271:⁄
226:⁄
98:scholar
3310:Sudoku
3159:Issues
3115:Brazil
3058:Quebec
3045:Mexico
3019:Turkey
3014:Russia
3009:Norway
2999:France
2987:Europe
2973:Turkey
2963:Taiwan
2953:Russia
2946:Manila
2873:Angola
2866:Africa
2758:Racino
2736:Casino
2729:Venues
2674:
2546:
2451:
1565:plus 1
1410:Patent
1394:Patent
1386:Patent
1375:Patent
1345:Patent
1316:Patent
1309:Patent
1294:Trixie
1278:Yankee
1270:Trixie
1259:Double
1255:Yankee
1251:Trixie
1219:Yankee
1215:Trixie
1186:Yankee
1179:Yankee
1155:Trixie
1148:Trixie
1061:treble
1055:Treble
966:double
960:Double
896:single
773:Yankee
587:treble
583:double
579:bettor
561:, and
366:Home:
139:ledger
100:
93:
86:
79:
71:
3127:Games
3004:Italy
2931:Japan
2926:India
2919:Macau
2909:China
2479:. 1.
2278:+ 1)(
2225:+ 1)(
2221:+ 1)(
2152:+ 1)(
2148:+ 1)(
2144:+ 1)(
2021:+ 1)(
1960:+ 1)(
1956:+ 1)(
1749:twice
1684:+ 1)(
1680:+ 1)(
1676:+ 1)(
1654:final
1597:+ 1)(
1593:+ 1)(
1499:+ 1)(
1495:+ 1)(
1367:Lucky
1230:Heinz
874:total
865:place
505:these
499:(20%)
486:(40%)
473:(60%)
393:(50%)
368:Evens
105:JSTOR
91:books
2892:Asia
2741:List
2672:ISBN
2544:ISBN
2449:ISSN
2303:+ 4(
2092:and
2046:+ 3(
1914:and
1856:and
1791:and
1789:Flag
1559:and
1547:and
1487:and
1253:and
1239:and
878:down
787:and
732:and
621:fair
615:and
607:and
445:fair
360:true
163:odds
157:and
77:news
2531:",
2520:",
1807:If
1537:abc
1505:abc
1435:If
1382:not
1266:not
861:win
659:or
589:or
429:(16
406:(33
196:to
192:or
125:In
60:by
3332::
3068:BC
2584:,
2574:,
2570:,
2498:.
2475:.
2461:^
2445:32
2443:.
2439:.
2396:^
2362:^
2325:3(
2317:OM
2307:+
2301:ab
2298:=
2296:OM
2286:+
2268:+
2260:+
2246:OM
2237:+
2233:+
2199:+
2195:+
2177:OM
2168:+
2164:+
2160:+
2138:=
2132:+
2128:+
2124:+
2102:OM
2088:,
2084:,
2068:2(
2060:OM
2050:+
2044:ab
2041:=
2039:OM
2029:+
2003:+
1995:+
1981:OM
1972:+
1968:+
1946:+
1942:+
1924:OM
1910:,
1891:OM
1882:+
1878:2(
1866:OM
1829:OM
1819:,
1815:,
1811:,
1700:+
1696:+
1692:+
1670:=
1668:OM
1661:OM
1587:=
1585:OM
1581:OM
1563:)
1555:,
1549:bc
1545:ac
1543:,
1541:ab
1533:OM
1527:+
1523:+
1519:+
1517:bc
1515:+
1513:ac
1511:+
1509:ab
1507:+
1483:,
1457:OM
1447:,
1443:,
1439:,
1430:OM
1416:.
1359:,
1353:,
1347:,
1233:,
1227:,
1221:,
1217:,
831:×
808:,
781:,
708:11
698:11
688:11
661:BD
657:BC
655:,
653:AD
651:,
649:AC
641:11
634:+
631:11
585:,
557:,
439:%)
416:%)
350:.
320:)/
313:10
277:+
220:+
173:.
3301:)
3297:(
3258:e
3251:t
3244:v
2705:e
2698:t
2691:v
2678:.
2666:"
2550:.
2538:"
2527:"
2516:"
2483:.
2481:9
2455:.
2327:a
2321:a
2309:b
2305:a
2288:b
2284:a
2280:b
2276:a
2274:(
2270:b
2266:a
2262:b
2258:a
2254:b
2250:a
2239:c
2235:b
2231:a
2227:c
2223:b
2219:a
2217:(
2213:c
2209:b
2205:a
2201:c
2197:b
2193:a
2189:c
2185:b
2181:a
2170:d
2166:c
2162:b
2158:a
2154:d
2150:c
2146:b
2142:a
2140:(
2134:d
2130:c
2126:b
2122:a
2118:d
2114:c
2110:b
2106:a
2094:d
2090:c
2086:b
2082:a
2070:a
2064:a
2052:b
2048:a
2031:b
2027:a
2023:b
2019:a
2017:(
2013:b
2009:a
2005:b
2001:a
1997:b
1993:a
1989:b
1985:a
1974:c
1970:b
1966:a
1962:c
1958:b
1954:a
1952:(
1948:c
1944:b
1940:a
1936:c
1932:b
1928:a
1916:c
1912:b
1908:a
1896:a
1884:b
1880:a
1874:b
1870:a
1858:b
1854:a
1841:c
1837:b
1833:a
1821:d
1817:c
1813:b
1809:a
1704:)
1702:d
1698:c
1694:b
1690:a
1686:d
1682:c
1678:b
1674:a
1672:(
1599:c
1595:b
1591:a
1589:(
1577:c
1573:b
1569:a
1561:c
1557:b
1553:a
1529:c
1525:b
1521:a
1501:c
1497:b
1493:a
1489:c
1485:b
1481:a
1469:c
1465:b
1461:a
1449:d
1445:c
1441:b
1437:a
1117:5
1113:1
1107:4
1103:1
1097:5
1093:1
1087:4
1083:1
1022:"
1018:"
996:"
992:"
985:4
981:1
975:5
971:1
918:5
914:1
844:i
840:B
836:2
833:B
829:1
826:B
821:i
817:B
813:2
810:B
806:1
803:B
799:i
792:,
765:D
762:O
758:2
755:B
751:1
748:B
744:D
741:O
737:2
734:B
730:1
727:B
723:D
720:O
712:6
705:×
702:6
692:6
670:4
666:1
637:6
627:6
617:D
613:C
609:B
605:A
535:3
531:2
496:5
492:1
483:5
479:2
470:5
466:3
436:3
432:2
426:6
422:1
413:3
409:1
403:3
399:1
390:2
386:1
346:x
341:1
336:D
331:x
327:1
322:x
318:x
309:4
299:4
292:D
287:1
281:)
279:b
275:a
273:(
268:b
262:D
258:b
254:a
247:D
243:a
239:b
235:a
229:b
224:)
222:a
218:b
216:(
211:D
206:b
202:a
198:b
194:a
190:b
188:/
186:a
184:(
182:b
178:a
116:)
112:(
102:·
95:·
88:·
81:·
64:.
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
