################ R Simple Simulation ################


################## Rolling a die 10 times ##################
vals=c(0,1)
counts=c(4,2)
urn=rep(vals,times=counts)
urn

sample(urn,size=10,replace=TRUE)
sample(urn,size=10,replace=TRUE)
sample(urn,size=10,replace=TRUE)
sample(urn,size=10,replace=TRUE)

mean(sample(urn,size=10,replace=TRUE))
mean(sample(urn,size=10,replace=TRUE))
mean(sample(urn,size=10,replace=TRUE))
mean(sample(urn,size=10,replace=TRUE))

results=replicate(10000,mean(sample(urn,size=10,replace=TRUE)))

head(results)
results[4000]
results[131]

mean(results)

sd(results)

### to make the plot
plot(table(results),type="h",ylab="Counts")

sum(results>=.8)




################## Polio Vaccine ##################
vals=c(0,1)
counts=c(399802,198)
urn=rep(vals,times=counts)
results=replicate(10000,sum(sample(urn,200000,replace=FALSE)))
sum(results<=56)


################## Calcium Supplement ##################
calcium=c(7,-4,18,17,-3,-5,1,10,11,-2)
placebo=c(-1,12,-1,-3,3,-5,5,2,-11,-1,-3)

urn=c(calcium,placebo)
urn

results=replicate(10000,mean(sample(urn,size=10,replace=FALSE)))
mean(results>=mean(calcium))
hist(results)