At sufficient scale, even incredibly unlikely things become quite probable.
runs <- 10000
x <- vector(mode = "numeric", length = runs)
for (i in 1:runs){
while (sum(sample(1:6, size = 3, replace = TRUE)) != 18){
x[i] <- x[i] + 1
}
}
summary(x)
quantile(x, c(0.5, 0.8, 0.9))
> summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0 62.0 149.0 216.2 300.0 1902.0
> quantile(x, c(0.5, 0.8, 0.9))
50% 80% 90%
149 350 495
The numbers very much agree with you. The median is 149. The 90th is 495 in the simulation, which is close enough to 496. There is very much a long tail in the data. So, the median and the average will not be the same. Is it a coincidence that mean is a 216?
Iteration counts gathered with Python and a (manual) binary search (actually faster than writing code).
runs <- 100000
x <- vector(mode = "numeric", length = runs)
for (i in 1:runs){
while (sum(sample(1:8, size = 3, replace = TRUE)) != 24){
x[i] <- x[i] + 1
}
}
summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0 146.0 353.0 511.8 708.0 5112.0
quantile(x, c(0.5, 0.8, 0.9))
50% 80% 90%
353 824 1187