Laplace’s rule of succession estimates the next success probability after independent success-or-failure trials: p = (s + 1) / (n + 2), where s is prior successes and n is completed trials. It is useful when few observations make the conventional s / n estimate misleadingly absolute. section 16.0
The added 1 and 2 preserve room for the opposite outcome. After five successes in five trials, the estimate is 6 / 7, or about 86%, rather than 100%; after five failures, it is 1 / 7, or about 14%. section 16.0
When successes equal trials, the resulting probability curve has a distinctive shape:
- It rises as the run of successes grows. section 16.3
- It rises at a decreasing rate rather than linearly. section 16.3
- It never reaches 100%, regardless of how many prior successes occurred. section 16.3
df %>%
mutate(laplace = (prior_successes + 1) /
(independent_trials + 2) * 100) -> df
print(df)
## prior_successes independent_trials laplace
## 1 1 1 66.66667
## 2 2 2 75.00000
## 3 3 3 80.00000
## 4 4 4 83.33333
## 5 5 5 85.71429
## 6 6 6 87.50000
## 7 7 7 88.88889
## 8 8 8 90.00000
## 9 9 9 90.90909
## 10 10 10 91.66667
## 11 11 11 92.30769
## 12 12 12 92.85714
## 13 13 13 93.33333
## 14 14 14 93.75000
## 15 15 15 94.11765
## 16 16 16 94.44444
## 17 17 17 94.73684
## 18 18 18 95.00000
## 19 19 19 95.23810
## 20 20 20 95.45455
Create a Laplace percentage column in R by applying the formula to prior successes and independent trials, then multiply by 100. section 16.3
Testing a free-throw streak
For a player’s free throws, separate attempts by game, calculate consecutive prior makes, and compare the next-shot percentages by streak length. Small counts at long streaks can produce apparent 100% rates, while simulated coin flips also contain streaks. The observed player-level pattern does not consistently rise with each additional make, supporting randomness rather than a hot hand. section 16.4.1 section 16.6
