Why the best team doesn't always win
Tonight's idea
Say team A beats team B with probability 0.6 in any single game. That is a big edge in real sport. Now ask how often A wins a series.
One game: 60%. Best of three: A needs two wins. The ways are WW, WLW, LWW. Each has probability \(0.6 \times 0.6 = 0.36\), \(0.6 \times 0.4 \times 0.6 = 0.144\), \(0.4 \times 0.6 \times 0.6 = 0.144\). Total 0.648, so 64.8%. Best of five: 68.3%. Best of seven: 71.0%. Longer series help the better team, but slowly, and never to certainty. Even a best-of-seven leaves the worse team winning 29 times in 100. That is what "probability 0.6" means (see Dice don't lie).
Tournament shape matters too. In a knockout (single elimination), one bad day and you are gone, so upsets pile up. In a round robin, where everyone plays everyone, luck has many games to average out, which is why league champions are usually stronger than cup winners. A round robin of \(n\) teams needs one game for every pair of teams, which is the number of ways to choose 2 teams from \(n\), written \(\binom{n}{2}\) (see How many ways). It gets expensive fast.
Then there are streaks. A player scores five in a row and the commentator says they are "hot". But flip a fair coin 20 times and the chance of seeing at least one run of four heads or four tails is about 77%. Runs are what randomness looks like. Researchers still argue about whether the hot hand in basketball exists; if it does, it is small.
Finally, averages: a batter can beat a rival in both halves of a season and still lose over the whole season, the reversal from Lying with averages. Draw it as on that sheet, one circle per at-bat, and the piles tell the story: the whole-season rate is a weighted average of the two halves, and the player with the big pile in the weak half loses it.
Watch one: best of seven, sweep
What is the chance the 0.6 team wins a best-of-seven four games to nothing? Four wins in a row: \(0.6 \times 0.6 \times 0.6 \times 0.6 = 0.6^4 = 0.1296\), about 13%. A sweep by the worse team is \(0.4^4 = 0.0256\), about 2.6%. So sweeps by the underdog happen roughly one series in forty.
Your turn
- Team A wins each game with probability 0.7. Compute the chance A wins a best-of-three by listing the cases WW, WLW, LWW. Compare with the single-game chance.
- Eight teams play a round robin (everyone plays everyone once). How many games is that? Explain where the number comes from, either by counting pairs or using Pascal's triangle.
- A knockout cup starts with 64 teams. How many matches are played in total? Give a one-sentence reason that works without drawing the bracket.
- In 20 coin flips there is roughly a 77% chance of a run of at least four the same. A coach sees a player miss four shots in a row and benches them "because they are cold". Write two sentences the team's statistician might say to the coach. Use the 77% number.
Stretch: how often does it go to game seven?
Two perfectly equal teams play a best-of-seven. The series lasts exactly seven games when, after six games, it is 3–3. How many of the \(2^6 = 64\) possible six-game sequences have exactly three wins for each side? (Pascal's triangle, row 6.) So what fraction of evenly matched series go the distance?
Real data from several sports show more game-sevens than this. Suggest two explanations that are not "the teams are equal", and one way you might test which is right.
Answers
- WW: \(0.49\). WLW: \(0.7 \times 0.3 \times 0.7 = 0.147\). LWW: \(0.147\). Total \(0.784\), so 78.4%, up from 70% for a single game.
- \(\binom{8}{2} = 28\) games. Each of 8 teams plays 7 others, \(8 \times 7 = 56\), but that counts every game twice, so 28. It is the third entry of row 8 of Pascal's triangle.
- 63 matches. Every match eliminates exactly one team, and 63 teams must be eliminated to leave one champion.
- Something like: "If this shooter's chance of scoring is known and stays the same on every independent shot, four misses do not change the chance of scoring next time. A run of misses alone does not prove the player has become cold; we would need to compare it with what their usual scoring rate predicts." Poorer shooters will have runs of misses more often, so this rule would not bench all players equally often.
Stretch. Row 6 of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1, so 20 of the 64 sequences are 3–3: \(20/64 = 31.25\%\). Possible reasons for more game-sevens in reality: home advantage alternates and the team behind is often at home in game six; a team that is behind tries harder or changes tactics; coaches rest players when ahead. To test: compare how often the trailing team wins game six at home versus away, or check whether the pattern persists in sports with no home advantage.