This article is from the Table Tennis (Ping Pong) FAQ, by ttennis@bu.edu with numerous contributions by others.

Eleven points, of course. A more precise question: Is a 4 out of 7 match of

11 point games the same as a 2 out of 3 match of 21 point games? Why do we

care? Over the last few years many tournaments both in the US and in other

countries have experimented with 11 point games to see if they make the

matches more exciting. Why don't you try such an event at your next

tournament? The results can still count for rating points (check with the

rating chairman for the current policy).

How do we measure the length of a match other than simply counting the

total points? The key is to realize that the length of a match is reflected

in the probability that the better player will lose. The longer the match,

the smaller the probability of an upset. Using standard modeling

assumptions (probability of winning a point is independent of the score) we

may relate the probability of winning a point to the probability of winning

a match under various formats. For simplicity, we will assume the

probability of winning a point does not depend on who serves. (It is

possible to take into account the dependence on who is serving, but the

conclusions remain the same.)

The table gives the probabilities of winning a match under various formats.

Each row of the table corresponds to a different format. For example, the

first row is for one game to 11 points. The "Games" column gives the number

of games you need to win the match, so "2" means a 2 out of 3 match. The

last row, labeled "2 sets" is for the tennis format: Each game is to 4

points with deuce at 3, each set is to 6 games with deuce at 5, and the

match is 2 out of 3 sets. I've used the old tennis format: no tie-breakers.

Note that I've also included a format of one game to 51. This is a popular

format for handicap matches.

Each column gives the probability of winning the match for a different

probability of winning a point. Note that the first column is the same for

all formats because it corresponds to a probability of winning a point of

0.5. If the two players are evenly matched and the format is fair (and all

these formats are), then the probability of winning the match is 0.5

regardless of the length. The larger the numbers in a given row, the longer

the match. The rows are in order with the shortest format at the top and

the longest format at the bottom.

So what can we conclude? A normal 2 out of 3 match is half way between the

11 point game formats of 3 out of 5 and 4 out of 7. It is slightly closer

to the 4 out of 7 format. A normal 3 out of 5 match is between the 11 point

formats of 5 out of 9 and 6 out of 11, but is closer to the 5 out of 9. The

51 point game is almost the same as a normal 2 out of 3. And finally, the

tennis format of 2 out of 3 sets is longer than all the other formats.

3.1.1 table PROBABILITY OF WINNING MATCH ------------------------------------------------- Format | Probability of Winning Point Points Games | 0.50 0.52 0.54 0.56 0.58 0.60 ------------------------------------------------- 11 1 | 0.50 0.58 0.65 0.72 0.78 0.84 21 1 | 0.50 0.60 0.70 0.79 0.86 0.91 11 2 | 0.50 0.61 0.72 0.81 0.88 0.93 11 3 | 0.50 0.64 0.77 0.86 0.93 0.97 21 2 | 0.50 0.65 0.79 0.88 0.94 0.98 51 1 | 0.50 0.66 0.79 0.89 0.95 0.98 11 4 | 0.50 0.66 0.80 0.90 0.96 0.98 11 5 | 0.50 0.68 0.83 0.92 0.97 0.99 21 3 | 0.50 0.69 0.84 0.93 0.98 0.99 11 6 | 0.50 0.70 0.85 0.94 0.98 1.00 2 sets | 0.50 0.71 0.87 0.95 0.99 1.00 -------------------------------------------------

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