In badminton, if you win the first game, how likely are you to win the match?

• A badminton match is best of 3 games, and each game goes to 21 points.
• Winning game 1 means you only need one more game, while your opponent needs two.
• Fans and commentators often say “the first game is everything.” Is that true?

Source: BWF World Tour singles matches, 2018 to early 2021 (Kaggle: Badminton BWF World Tour), scraped from the official BWF website.

ms = read.csv("ms.csv")   # men's singles
ws = read.csv("ws.csv")   # women's singles

matches = rbind(ms, ws)                        
matches = subset(matches, retired == "False")   
nrow(matches)

[1] 6624

head(matches[, c("team_one_players", "team_two_players",
                 "game_1_score", "winner")])

   team_one_players        team_two_players game_1_score winner
1     Jun Hao Leong Kantawat Leelavechabutr        11-21      1
2      Teck Zhi Soo  Chico Aura Dwi Wardoyo         21-7      1
3     Mek Narongrit           Yee Han Chong        21-16      1
4  Mithun Manjunath     Henrikho Kho Wibowo        21-17      1
5        Chirag Sen         Hashiru Shimono        13-21      2
6 Shreyansh Jaiswal   Fikri Ihsandi Hadmadi        19-21      2

Let \(p\) be the true probability that the player who wins game 1 goes on to win the match.

We estimate \(p\) with the sample proportion:

\[ \hat{p} = \frac{x}{n} = \frac{\text{matches won by the game 1 winner}}{\text{total matches}} \]

For our data:

\[ \hat{p} = \frac{5442}{6624} = 0.822 \]

So the player who won game 1 went on to win the match 82.2% of the time.

If game 1 did not matter at all, we would expect \(p = 0.5\).

Out of 6624 matches, only 1182 were comebacks. That’s about 1 in 6 matches.

Our \(\hat{p}\) comes from a sample, so we give a range of believable values for the true \(p\):

\[ \hat{p} \pm z_{0.025} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hspace{1 cm} z_{0.025} = 1.96 \]

\[ 0.822 \pm 1.96 \sqrt{\frac{0.822(1-0.822)}{6624}} = (0.812, \; 0.831) \]

p_hat = x / n
se = sqrt(p_hat * (1 - p_hat) / n)
c(lower = p_hat - 1.96 * se, upper = p_hat + 1.96 * se)

    lower     upper 
0.8123373 0.8307787 

We are 95% confident that the game 1 winner wins between 81.2% and 83.1% of matches.

\[ H_0: p = 0.5 \quad \text{(game 1 doesn't matter)} \qquad H_a: p > 0.5 \quad \text{(game 1 winner has the edge)} \]

\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.822 - 0.5}{\sqrt{\frac{0.5(0.5)}{6624}}} = 52.3 \]

prop.test(x, n, p = 0.5, alternative = "greater", correct = FALSE)

    1-sample proportions test without continuity correction

data:  x out of n, null probability 0.5
X-squared = 2739.7, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is greater than 0.5
95 percent confidence interval:
 0.813689 1.000000
sample estimates:
       p 
0.821558 

A \(z\) of 52.3 is large and the p-value is essentially 0, so we reject \(H_0\). Winning game 1 is a real advantage, not luck.

Winning game 1 by 2 points (like 21-19) still leaves about a 1 in 4 chance of losing. Winning it by 11+ (like 21-8) is almost a guaranteed win. The dashed line is 50%.

This interactive plot shows every game 1 margin, not just 4 groups. Hover over a point to see the exact chance and how many matches had that margin. The line climbs steadily: every extra point won in game 1 makes the lead safer.

men   = subset(matches, discipline == "MS")
women = subset(matches, discipline == "WS")

prop.test(sum(men$won_after), nrow(men), correct = FALSE)$estimate

        p 
0.8089949 

prop.test(sum(women$won_after), nrow(women), correct = FALSE)$estimate

        p 
0.8373679 

• Men: game 1 winner wins 80.9% of the time, 95% CI (79.6%, 82.1%)
• Women: game 1 winner wins 83.7% of the time, 95% CI (82.4%, 85%)

Both are far above 50%. Comebacks are slightly less common in women’s singles.

• Game 1 matters a lot. The player who wins it goes on to win the match about 82% of the time, and we are 95% confident the true value is between 81.2% and 83.1%.
• It is not a coin flip: the hypothesis test strongly rejects \(p = 0.5\).
• How you win game 1 matters too: a close 21-19 win holds up about 73% of the time, while a big 11+ point win holds up 93% of the time.
• The pattern is the same for men and women.

Limitations

• This is association, not causation: the better player often wins game 1 and the match.
• Only World Tour singles from 2018–2021, and matches with injury retirements were removed.

Big takeaway: a fast start is a huge advantage in badminton, but about 1 in 6 matches still gets turned around. Never give up after game 1!