library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

lets create 100 numbers due to 0.1 probability and 100 trials ( suppose that we make a lottery trial ) with binomial dist

numbers <- rbinom(100, 100, 0.1)

numbers

##   [1]  8 12 13 12 14 14 12  8 11 17  8 11  8  9 11  9 11 15  8  6  8 11  6
##  [24] 10  9 11  8 14 10 11 14 12 12 11 17  6  9 10 14 13 10  8 12  9  8  9
##  [47] 13  8  7 10 12 10  7  8 15 11  8 15  7 11 11  4 17  6 15 10 13  8  8
##  [70]  4 11 10  8 13 11  8 10 13 11 10  9 15 11  9 13 11  9  7 10  5  9  7
##  [93] 10  7 10  8 10  9 13 12

function to calculate probabilities of each value due to binomial distribution

binomial_dist <- function(n,p,x) {
  temp_C <- (factorial(n) / (factorial(x) * factorial(n-x)))*(p^x)*((1-p)^(n-x))
  return(temp_C)
}

an example

binomial_dist(100,0.1,2)

## [1] 0.001623197

lets calculate each of our observations binomial probabilities with our function

binom_df<-NULL
for (i in 1:100){
  binom_df[i] <- binomial_dist(100,0.1,numbers[i])
}

make it data frame

full_df <- data.frame(numbers,binom_df)

head(full_df)

##   numbers   binom_df
## 1       8 0.11482303
## 2      12 0.09878801
## 3      13 0.07430209
## 4      12 0.09878801
## 5      14 0.05130383
## 6      14 0.05130383

and plot all of the probabilities !

plot(numbers,binom_df,type="h",main="Binom Distribution",ylab="Binomial Distributions of Values", xlab = "Number of Trials",col="blue")

So as expected, we created 100 treatments with 0.1 occuring probability and as seen in the graph, the probability gets top point between 9 and 11 treatments