OxfordCroquet Logo

Dr Ian Plummer

Donation Button

http://www.oxfordcroquet.com/tech/carter1/index.asp
Technical
The Maths of Matches

Kevin Carter shows that best-of-3 and best-of-5 matches have only a small effect on who wins a match over a single game. Kevin's results are discussed here.

It is generally considered that in top flight events it is much fairer to have best-of-three matches than single games, and even better to have best-of-five. Actually, the effect of having matches of up to three or five games is not as great as most people think.

It is certainly correct to try to reduce the effect of luck or random events in a game, and there is nothing worse than a ‘sterile’ game. At advanced level this means that Player A gets in and goes to 4-back, Player B misses a 19-yard shot and Player A completes the game with a TP. This is common when conditions are easy - a holding surface which is easy to judge, along with hoops which ‘give’.

So, by how much is the probability of the superior player winning improved by playing longer matches?

Let us consider first a best-of-3, where Player A has a 60% chance of winning each game. There are three ways in which he can win the match:

  1. Winning the first two games (which we shall notate as ‘AA’).
  2. Losing the second (ABA)
  3. Losing the first (BAA)

The Probability of AA is 0.6 x 0.6 = 0.36; ABA is 0.6 x 0.4 x 0.6 = 0.144; and BAA is 0.4 x 0.6 x 0.6 = 0.144. The total of these three is 0.65. So, this means that if the chance of winning a single game is 60%, then the chance of winning a best-of-3 is 65% - hardly a transformation! Of course, this ignores any psychological effects and also whether one player would tire more than the other.

If you repeat the calculation for when player A has a 70% chance in each game then his probability of winning a best-of-3 is improved to 78% - again, an improvement, but the substantially better player is still far from a certainty when playing a best-of-3.

Now looking at best-of-5, Player A can win in any of the following ways:

AAA

0.63

BAAA, ABAA, AABA

0.63 x 0.4 x 3

BBAAA, BABAA, BAABA, ABABA, ABABA, AABBA

0.63 x 0.42 x 6

The figures to the right show the calculations. When they are worked out and totalled we come to 68% - so, adding a further two games to a match pushes up the chance of the 60% player winning from 65% to just 68%.

For a single game 70% we improve our 78% for best-of-3 to 84% for best-of-5. This is a little more impressive; we have almost halved the chance of a ‘rogue result’.

So, to summarise:

Probability of winning a single game

Probability of winning Bo3

Probability of winning Bo5

60%

65%

68%

70%

78%

84%

n%

n2 + 2n2(n-1)

n3+3n3(n-1)+6n3(n-1)2

The last row above gives the general formula, which allows anybody to calculate probabilities for Bo3 and Bo5 for other values.

The next question is whether it is worthwhile increasing the length of matches to three or five games. A 32-player knock-out can be comfortably completed in two days, with just five single-game rounds. However, even top players struggle to complete five best-of-3s in three days, and B-class players or seniors would take four. So, you are trading a slightly fairer result against the length of the tournament increasing by 50-100%.

Is there another solution? Yes - more challenging conditions. If our player A has a 60% chance of beating B in easy conditions then his chance in a more interactive game might be, say, 70%. He can bring to bear his superior skills at making and maintaining breaks, while the chance of his slightly inferior opponent getting the first break and winning a sterile game is reduced.

If you accept this very reasonably premise then there is a surprising conclusion: that a single game in challenging conditions (70%) is more likely to go the superior player than the best-of-5 in easy conditions (68%)!

Author: Kevin Carter
All rights reserved © 2017-2017


Updated 30.iii.17
About, Feedback
www.oxfordcroquet.com/tech/carter1/index.asp
on www.oxfordcroquet.com
Hits: 298