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title: “Assignment1” author: “Sabuj Ganguly” date: “2025-09-05” output: html_document

##Coin Flip Simulation Analysis:

rm(list=ls())

n=5
p=0.5  #assuming the coin is unbiased

i=0:n
theo_prob=dbinom(i,n,p) #the given problem is the binomial distribution of size 5 and probability 0.5
theo_prob

## [1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125

set.seed(123)
sim_no=1000  #no. of repetition of the coin flipping experiment is 1000

single_experiment=sample(c(0,1),n,replace=TRUE) #this simulates one experiment of flipping a coin n=5 times
single_experiment

## [1] 0 0 0 1 0

sum(single_experiment) #this sum provides us the no. of heads we are getting in one particular experiment

## [1] 1

flips=replicate(sim_no,sum(sample(c(0,1),n,replace=TRUE))) #replicating it 1000 times will provide us the no. of heads occurred in 1000 repetition

table(flips)  #this table provides how many times each possible no. of heads (0,1,2,3,4,5)occurred in the 1000 experiments 

## flips
##   0   1   2   3   4   5 
##  32 148 328 300 160  32

simulated=table(flips)/sim_no
simulated

## flips
##     0     1     2     3     4     5 
## 0.032 0.148 0.328 0.300 0.160 0.032

results=data.frame(Heads=0:5,Theoretical=round(theo_prob,3),Empirical=round(as.numeric(simulated[as.character(0:5)]),3)) #comparison of theoretical and empirical result
print(results)

##   Heads Theoretical Empirical
## 1     0       0.031     0.032
## 2     1       0.156     0.148
## 3     2       0.312     0.328
## 4     3       0.312     0.300
## 5     4       0.156     0.160
## 6     5       0.031     0.032