The Binomial Distribution in Shooting

KEYWORDS:Teaching;Double trap competition;Prediction.

Miltiadis S. ChalikiasUniversity of Central Greece, Laima, Greece.e-amil: mchalikias@ucg.gr

SummaryThe binomial distribution is used to predict thewinner of the 49th International Shooting SportFederation World Championship in double trapshooting held in 2006 in Zagreb, Croatia. Theoutcome of the competition was definitelyunexpected.test_344 87..89

INTRODUCTION

This article describes a shooting competitionappropriate for illustrating the use of thebinomial distribution with students. Sportingapplications often keep student interest at a highlevel. As will be seen, probability theory may beused to predict the winner of the shooting compe-tition at any time during the competition. Suchprediction, in addition to being of pedagogical andtechnical interest, is useful for someone who iswilling to bet on the result of the competition. It iswidely known that betting has infiltrated sports.Numerous companies render betting possible onalmost all sports. Two of the most common betsoffered by such companies are the prediction ofthe winner and the prediction of the exact score.Each of these is possible using a binomial model,but we only deal with estimating the chance ofwinning in this article.

In particular, this article deals with a double trapcompetition. In the preliminaries of this competi-tion, each shooter fires three fifties of shots. Everyshot at a clay target is either successful (hits thetarget) or unsuccessful (does not hit the target).The top six shooters of the preliminaries move on tothe finals and compete with one another. In thefinals they fire 50 shots and their results are addedto the results of the preliminaries. Thus, forinstance, if a shooter has 144 successful shots out of150 in the preliminaries and 47 in the finals, hisoverall score is 191.

BINOMIAL DISTRIBUTION

By way of reminder, when the probability of a suc-cessful shot against a target is p and that of anunsuccessful shot is 1 p, the probability of havingx successful shots in n attempts is given by

f xn

xp px n x( ) = ( ) 1

for x = 0, 1, . . . , n. Consequently the probability ofno more than x successful shots in n attempts isgiven by

F xn

kp p

k

xk n k( ) = ( )

=

0

1

THE2006WORLDCHAMPIONSHIP

We consider the 49th International Shooting SportFederation (ISSF) World Championship Competi-tion held in Zagreb, Croatia, in 2006. Full detailsare available via the ISSFs Web page (http://www.issf-shooting.org/). In particular, the finalresults of each shooter in both the preliminariesand the finals are given at this site as well as theresults of each of the 50 final shots. We will seethat in this competition the Russian shooter VitalyFokeev won in an extraordinary way despite thefact that he held the lowest position of the finalistsafter the preliminaries. The six shooters in thefinals and their performance in the preliminariesare given in figure 1.

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Teaching Statistics. Volume 31, Number 3, September 2009 87

We now evaluate the chance the Russian shooterwins using information from the preliminaries. Todo so, let X1 be the number of successful shots inthe finals by Fokeev, X2 that by Hu, X3 that byWang, X4 that by Gerebics, X5 that by Faulds andX6 that by Richmond. In the calculations below,we presume that all shooters fire more than 40 suc-cessful shots in the finals. That is, we assumeXi > 40, for i = 1, 2, 3, 4, 5, 6. As may seen in thecalculations below, there is little loss in accuracyby doing so.

The shooters performance in the preliminariesgives estimates of the probabilities of success in thefinals. In particular, we get 140/150 = 0.933 forFokeev, 141/150 = 0.940 for Richmond and 142/150 = 0.947 for the remaining four shooters. In thefollowing calculations we will adopt the assumptionthat the outcomes are independent and the prob-ability of success is fixed for each shooter. The prob-ability P of Fokeev winning the competition, giventhe situation after the preliminaries, may then beestimated as

P P X P X P X P X P X P X

P X

= =( ) ( ) ( ) ( ) ( ) ( )+ =(

1 2 3 4 5 6

1

50 47 47 47 47 48

49)) ( ) ( ) ( ) ( ) ( )+ =( ) (

P X P X P X P X P X

P X P X2 3 4 5 6

1 2

46 46 46 46 47

48 45)) ( ) ( ) ( ) ( )+ =( ) ( ) (

P X P X P X P X

P X P X P X3 4 5 6

1 2 3

45 45 45 46

47 44 44)) ( ) ( ) ( )+ =( ) ( ) ( ) (

P X P X P X

P X P X P X P X4 5 6

1 2 3 4

44 44 45

46 43 43 43)) ( ) ( )+ =( ) ( ) ( ) ( ) (

P X P X

P X P X P X P X P X5 6

1 2 3 4 5

43 44

45 42 42 42 42)) ( )+ =( ) ( ) ( ) ( ) ( ) (

P X

P X P X P X P X P X P X6

1 2 3 4 5 6

43

44 41 41 41 41 42))= + + + +

+

. . . . .

.

0 0016426 0 0003869 0 0000181 0 0000002 0 0000000

0 00000000 0 0000000 0 0020477+ =. .

Simply speaking, this is the likelihood of Fokeevachieving three successful shots more than Wang,Hu, Gerebics and Faulds (the shooters who had

two successful shots more than Fokeev in the pre-liminaries) and two successful shots more thanRichmond (the shooter who had achieved one suc-cessful shot more than Fokeev in the preliminar-ies). Some computation shows that P is less than1.0% and, in particular, is nearly equal toP = 0.0020477. Consequently, a betting companycould set the win return of the Russian shooterbefore the finals up to 1/P = 488.35. In otherwords, one could risk 1 on Fokeev before thefinals predicting his victory, in order to win487.35! Similar calculations give the odds ofwinning for the rest of the shooters; for instance, itis 15.46% for Wang, Hu, Faulds and Gerebics cor-responding with a return of 6.47 which is muchlower than that of the Russians.

The large difference of the returns clearly demon-strates the extent of surprise and accomplishment ofFokeev who was successful in the finals. As an addi-tional way of understanding this result, we can saythat if the finals were repeated 488 times, the victoryof the Russian shooter would be expected only oncewhereas Wangs, Hus, Fauldss and Gerebicsswould each be expected about 75 times.

We get a clear picture of the entire competition in thescoreboard displayed in figure 2 (also extracted viathe ISSF Web page: http://www.issf-shooting.org/).

Finally we evaluate the likelihood of success of theRussian shooter in his two final shots. The Russianshooter, by firing 48 consecutive successful shotsin the finals, held 188 successful shots overall andwas tied with Hu. The other four shooters had

Fig. 1. 2006 ISSF world championshippreliminaries

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88 Teaching Statistics. Volume 31, Number 3, September 2009

185 or fewer successful shots and could no longerhope for victory. The most likely outcome was forFokeev and Hu to each fire two successful shots;this may be estimated as having a chance of (140/150)2(142/150)2 = 78.1%. The chance Fokeev makeseach of his last two shots and Hu misses just his finalshot may be estimated as (140/150)2(142/150)(8/150) = 4.4%. We may also, of course, estimate thesechances using the results of the first 198 shots.Doing so, the estimated chance of success for eachshooter is 188/198 = 0.9494. Using this commonchance of success estimate, the above two estimatesare somewhat modified to (188/198)4 = 81.3% and(188/198)3(10/198) = 4.3%, respectively.

DISCUSSION

The outcome of the competition was definitelyunexpected. In the finals, however, what often playsa significant role is the psychological factor, whichcannot easily be quantified. It is remarkable that,while Fokeevs opponents performed worse thanthey did in the preliminaries, Fokeev himself, evenholding the lowest performance in the preliminaries,

was successful in each of his final 50 shots. Hisperformance could be due to his lack of stress sincehe probably had little hope for winning. Alterna-tively, someone could maintain that he is the kind ofshooter who can handle psychological pressure welland is able and specially trained to perform well incritical competitions. Of course we could onlyattempt to reach safe conclusions about such possi-bilities by examining a suitable quantity of histori-cal data in this context.

While beyond the scope of this paper, one mayestimate the chance of an individual winning thecompetition by using an adaptive procedure. In par-ticular, the probability of success need not be fixedfor any given shooter, but may be updated afterevery shot. Finally, the author wants to make itclear that the methodology in this paper is intendedfor pedagogical purposes only; it is not intended forbetting purposes and financial benefit!

Acknowledgments

The author would like to thank the ISSF News forpermission to include the two figures extracted fromits Web site.

Fig. 2. 2006 ISSF world championship final results

2009 The AuthorJournal compilation 2009 Teaching Statistics Trust

Teaching Statistics. Volume 31, Number 3, September 2009 89