The Pennant has been decided by a best-of-seven series since 1985, before which it was best-of-five. Last week, the Red Sox won the ALCS, but would have lost it if the series were still decided by best-of-five – that is, assuming the psychology would not have been different in a determinative way. This leads us to reflect on how much more accurate a best-of-seven criterion is, compared to best-of-5. “Accuracy” here means: the better team wins the series. And I know of no other objective basis upon which to say that team A is better than team B than this: after playing N games, the number of games that A wins is greater than the number that B wins. Applying the principles of probability, in the limit that N is very large, if the ratio of (the number of wins by team A) to (N) would converge to some number p, then we could say that p is the probability that A beats B in any game. From all this, it follows that A is “better” than B by definition if p is greater than 0.5 or 50%.
If the two teams are equally matched — that is, if there is a 50% chance of one team beating the other –, then there is also an equal chance of either team winning a series, regardless of the length of the series. Likewise, if one team had total domination over the other, such that one could say there is a 100% chance of victory in any game, then the probability of the better team winning is also 100% for any length series. So, the rationale for having a series at all, let alone a long series, must have to do with cases in between those two extreme cases.
In order to apply probability theory in a straightforward way, we must assume not only that (i) it is sensible to claim that there is a definite probability p that team A will beat team B, but also (ii) that the games are independent. Both of these assumptions are problematical. Assumption (i) suppresses the fact that each team changes its configuration (minimally: its pitcher) and thus also its game plan each game, so that one might think the probabilities shift around from game to game in reality; (ii) pretends that there is no such thing as “momentum,” that psychological spurt of triumphalism or defeatism than can affect the probable outcome of a subsequent game based on memory of the past.
Despite these objections, let us simply take these assumptions as hypothesis: and once given, it is straightforward to calculate the probabilities of winning a series. One must add up the probabilities for every possible sequence of games leading to winning M as defined for the series. M is now 3 for the Divisional series, and 4 for the Pennant and World Series. Adopting the perspective of the eventual winner for simplicity, the final game in each potential sequence is always a win (W). This reduces the length to one less than the actual number of games for calculating permutations, which helps a bit. A number of probabilities are multiplied together equal to the length of the sequence, of which M are always p; the probability of each loss is (1-p). Thus, for the best-of-5 category, for example, one possible sequence is:
LWWW
and this would be assigned probability (1-p) p3. Each possible sequence is mutually exclusive, and thus the total probability is equal to the sum of the probability of each sequence.
The results are shown in the graph. As predicted, the series probabilities all match for the case of 50% and 100% probability. For probabilities in between, the longer the series, the better chance the better team will win, as expected.
So, for example (see also table), if Team A has a 60% chance of beating Team B – which would be pretty good against another championship-quality team – the chance of winning the series rises from 60% for a single game to 65% for best-of-three, 68% for a best of five, and to 71% for best-of-seven.
| P(win) | 1 game | 2 of 3 | 3 of 5 | 4 of 7 |
| 50% | 50.0% | 50.0% | 50.0% | 50.0% |
| 55% | 55.0% | 57.5% | 59.3% | 60.8% |
| 60% | 60.0% | 64.8% | 68.3% | 71.0% |
| 70% | 70.0% | 78.4% | 83.7% | 87.4% |
| 80% | 80.0% | 89.6% | 94.2% | 96.7% |
| 80% | 80.0% | 89.6% | 94.2% | 96.7% |
| 90% | 90.0% | 97.2% | 99.1% | 99.7% |
This is rather astonishing. For p=0.6 in a thousand series, the expectation is that in only 27 more of them would the “right team” win by virtue of using a four-out-of-seven rule rather than three-out-of-five. What is noteworthy is that in 290 of those series, the “wrong team” is expected to win anyhow. Roughly a third of the time, the weaker team will take the series. And the reduction of the chance is rather inconsiderable going from a five to a seven-game series.
I conclude that there is a law of diminishing returns that is already reached by the time a series is stretched to best-of-five. The length of the series should be governed not by math, but solely by the fans’ appetite: how many games we want to watch.
(At least for the Pennant series, I vote to go back to the five-game sequence. It is enough.)
I have a sneaking suspicion that a vote taken by an impartial jury of aficionados as to which team is better would be as accurate as the outcome of the series. Except for one small problem of course: being the better team should, by definition, mean winning.
