Additionally, I tried to calculate the probability of occurrence of the U-tum with loss events. I calculated both of the case having the 25 and 45 samples. P[X=k] (p(1 - p)n-k = for k 0,1,2,...,n (k) =nck = n! k! (n-k)! When n=25, k=22, and p=1/3 EP[X]=25!*(1/3)*(2/3)/2213!=2.17*10* Also, when n=45, k= 32, and p=1/3 EP[X] 45!*(1/3)32*(2/3)13/32!13! 2.02*107 Furthermore, in case of determining the probability of "rarity" through simulation, I repeated 1000 trials of occurring the U-turn with loss events 45 times, and I collected data on the number of the events by using RStudio Desktop. > coin hist(coin,col="skyblue") Oraphic Device (ACT) 8 Ophic Devic Histogram of coin Histogram of chisqcoin 25 2 4 6 8 10 coin chiqcoin > coin m s c(m,s) [1] 14.760000 3.094184 > cs32 cs32 [1] 31.04433 > chisqcoin hist(chisqcoin,col="skyblue") > abline(v=cs32,col="red") > length(chisqcoin[chisqcoin> cs32])/1000