# How Much Is Winning a Matter of Luck? A Comparison of 3 × 3 and 5v5 Basketball

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

**H1:**

**H2:**

## 2. State of the Art of Explaining Winning Due to Luck

## 3. Methodology

_{i}is the expected outcome and AO

_{i}is the actual outcome of games for team i.

#### 3.1. Performance Indicator

_{i,j}is the performance indicator of team i in basketball form j, AO

_{i,j}is the actual outcome, EO

_{i,j}the expected outcome of games for team i in basketball form j, and k

_{i}is the total number of games of team I, k

_{i}was essential to weight the proportions of the actual and expected outcomes.

#### 3.2. Surprise Index

_{i,j}) between the two variables.

_{i,j}is 0 if there was no difference in the actual and the expected outcomes of the game and 1 if there was a difference, so the game ended with a surprise. With the new surprise variable, we were able to create a Surprise Index formula for basketball World Cups, which measures the ratio of games that ended with a surprise to all possible surprises.

#### 3.3. The Probit Models

^{2}s, the goodness-of-fit measures of the models [40]. The higher the magnitude of the pseudo R

^{2}is in a basketball form, the less luck is involved. The probit models were run separately for basketball forms and sexes.

#### 3.4. Relative Score Difference Index

## 4. Data

## 5. Results

#### 5.1. Performance Indicator and Surprise Index

_{2}). The standard deviation of the performance indicators was the highest for the 5v5 men’s basketball World Cup in 2014. Interestingly, the lowest value was zero. In the 3 × 3 women’s World Cup in 2019, every actual outcome was the same as our expectation based on the final outcomes. The Surprise Indexes showed analogous results but with different magnitudes (Table 3).

#### 5.2. The Probit Models

^{2}of the probit model was the lowest for this form (0.446). Conversely, the women’s 3 × 3 World Cups had the highest pseudo R

^{2}(0.719). Luck played the smallest role in these tournaments, and the expected outcomes of the games explained the actual outcomes in the women’s 3 × 3 World Cups quite accurately. All probit models and coefficients were significant at the 1% level (Table 5).

#### 5.3. Relative Score Difference Index

## 6. Discussion

_{1}and H

_{2}). On the one hand, we expected higher luck to be involved in 3 × 3 than in 5v5 basketball, and the ANOVA test of the RSDI clearly demonstrated that assumption. The specific characteristics of 3 × 3 games, for example, their intensity [13,15,16] and their outdoor courts, tend to increase the unpredictability of this form. We could not distinguish which characteristic accounts for the difference, but this was not within the scope of our study. On the other hand, higher luck was expected in the men’s competitions and this hypothesis was also obviously confirmed. The men’s competitions were more balanced (see the results of the RSDI), and this tended to lead to higher luck in the competitions [7]. This finding is in line with previous studies that have found greater competitive balance in men’s contests [23,24].

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Variable | M | SD | Min | Max |
---|---|---|---|---|

Year | 2010 | 2019 | ||

Form (0 = 5v5, 1 = 3 × 3) | 0.432 | 0.496 | 0 | 1 |

Sex (0 = men, 1 = women) | 0.411 | 0.492 | 0 | 1 |

Team A points | 51.503 | 31.977 | 4 | 129 |

Team B points | 46.593 | 30.861 | 2 | 119 |

Final ranking A | 8.668 | 6.539 | 1 | 32 |

Final ranking B | 11.611 | 7.005 | 1 | 32 |

Expected outcome | 0.679 | 0.467 | 0 | 1 |

Actual outcome | 0.673 | 0.470 | 0 | 1 |

PI | SI | Probit | RSDI | |
---|---|---|---|---|

PI | 1 | |||

SI | 0.998 | 1 | ||

Probit | −0.859 | −0.885 | 1 | |

RSDI | 0.685 | 0.724 | −0.960 | 1 |

**Table 3.**The standard deviations of the performance indicators, the Surprise Indexes, and the total number of games for teams in 3 × 3 and 5v5 basketball World Cups between 2010 and 2019.

Year | Form | Sex | SD | SI | N |
---|---|---|---|---|---|

2010 | 5v5 | Men | 0.066 | 0.050 | 160 |

2010 | 5v5 | Women | 0.062 | 0.081 | 124 |

2014 | 5v5 | Men | 0.105 | 0.158 | 152 |

2014 | 5v5 | Women | 0.067 | 0.056 | 72 |

2017 | 3 × 3 | Men | 0.072 | 0.063 | 96 |

2017 | 3 × 3 | Women | 0.027 | 0.021 | 96 |

2018 | 5v5 | Women | 0.085 | 0.094 | 64 |

2018 | 3 × 3 | Men | 0.078 | 0.083 | 96 |

2018 | 3 × 3 | Women | 0.077 | 0.125 | 96 |

2019 | 5v5 | Men | 0.042 | 0.043 | 184 |

2019 | 3 × 3 | Men | 0.091 | 0.125 | 96 |

2019 | 3 × 3 | Women | 0.000 | 0.000 | 96 |

**Table 4.**The total standard deviations of the performance indicators, the Surprise Indexes, and the total number of games for teams by basketball form and sex.

Form | Sex | SD | SI | N |
---|---|---|---|---|

3 × 3 | Women | 0.047 | 0.049 | 288 |

5v5 | Women | 0.072 | 0.077 | 260 |

5v5 | Men | 0.074 | 0.081 | 496 |

3 × 3 | Men | 0.081 | 0.090 | 288 |

**Table 5.**The results of the luck-measuring probit models based on the unexplained variance, separated by basketball form and sex.

Variable | (1) | (2) | (3) | (4) |
---|---|---|---|---|

Men’s 5v5 | Men’s 3 × 3 | Women’s 5v5 | Women’s 3 × 3 | |

AO | ||||

EO | 2.815 *** | 2.590 *** | 3.203 *** | 3.329 *** |

(0.251) | (0.360) | (0.437) | (0.378) | |

Constant | −1.426 *** | −1.221 *** | −0.992 *** | −1.314 *** |

(0.209) | (0.319) | (0.201) | (0.239) | |

Observations | 248 | 144 | 130 | 144 |

Pseudo R2 | 0.566 | 0.446 | 0.650 | 0.719 |

**Table 6.**The total standard deviations of the Relative Score Difference Indexes and the total number of games for teams by basketball form and sex.

Form | Sex | M | SD | N |
---|---|---|---|---|

5v5 | Women | 1.178 | 1.773 | 144 |

3 × 3 | Women | 1.531 | 2.002 | 144 |

5v5 | Men | 2.508 | 4.275 | 248 |

3 × 3 | Men | 4.303 | 6.816 | 130 |

**Table 7.**The results of the analysis of variance performed with the Relative Score Difference Indexes.

Source | df | F | p |
---|---|---|---|

Model | 3 | 15.15 | 0.000 |

Form (3 × 3, 5v5) | 1 | 37.48 | 0.000 |

Sex (women, men) | 1 | 10.29 | 0.001 |

Form * Sex | 1 | 4.63 | 0.032 |

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**MDPI and ACS Style**

Csurilla, G.; Boros, Z.; Fűrész, D.I.; Gyimesi, A.; Raab, M.; Sterbenz, T. How Much Is Winning a Matter of Luck? A Comparison of 3 × 3 and 5v5 Basketball. *Int. J. Environ. Res. Public Health* **2023**, *20*, 2911.
https://doi.org/10.3390/ijerph20042911

**AMA Style**

Csurilla G, Boros Z, Fűrész DI, Gyimesi A, Raab M, Sterbenz T. How Much Is Winning a Matter of Luck? A Comparison of 3 × 3 and 5v5 Basketball. *International Journal of Environmental Research and Public Health*. 2023; 20(4):2911.
https://doi.org/10.3390/ijerph20042911

**Chicago/Turabian Style**

Csurilla, Gergely, Zoltán Boros, Diána Ivett Fűrész, András Gyimesi, Markus Raab, and Tamás Sterbenz. 2023. "How Much Is Winning a Matter of Luck? A Comparison of 3 × 3 and 5v5 Basketball" *International Journal of Environmental Research and Public Health* 20, no. 4: 2911.
https://doi.org/10.3390/ijerph20042911