I wrote the value of probability p as $1/3,$ so$,$ the value should be $0\mathrm{.}333...$ You wrote it as $0\mathrm{.}13.$ it is a mistake isnt it$?$ Please revise the result again.$$ Additionally, I tried to calculate the probability of occurrence of the U$-$turn with loss events.

I calculated both of the case having the $25$ and $45$ samples.

$P[x=k]=([n$

$k])p^k(1-p{)}^{n-k}$

for $k=0,1,2,$dots,$n$

$([n$

$k])={?}_n{C}_k=\frac{n!}{k!(n-k)!}$

When $n=25,k=22,$ and $p=\frac{1}{3}$

$P[x]=25!**(\frac{1}{3}{)}^{22**}\frac{(\frac{2}{3}{)}^3}{22}!3{2\mathrm{.}17}^{**}{10}^{-8}$

Also, when $n=45,k=32,$ and $p=\frac{1}{3}$

$P[x]=45!**(\frac{1}{3}{)}^{32**}(\frac{2}{3}{)}^{\frac{13}{3}}!132\mathrm{.}02**{10}_{?}^{-7}$

$$ 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.

$>$ coinrbinom$(1000,45,0\mathrm{.}33$

$>$ coinrbinom$(1000,45,0\mathrm{.}33)$

mean$(coin)$

coin $)$

$>c(m,s)$

$[1]14\mathrm{.}760000,3\mathrm{.}094184$

$>cs32$

$[1]31\mathrm{.}04433$

$>$ chisqcoin coin$-m{)}^{{?}^{{?}^{?}}}\frac{2}{s^{{?}^{{?}^{?}}}}2$

$>$ hist$($chisqcoin$,$col$=$"skyblue"$)$

$>$ abline$(v=cs32,$col$=$ red"$)$

$>$ length$($chisqcoin$[$chisqcoin $cs32)\frac{?}{1000}$