# Problem: 
#    f friends call (at random) on d days.
#    Find the probability that at least one friend calls on each day.

# sim(f,d,nrep) uses a simulation to approximate the probability.
#    nrep is the number of simulation repetitions.

# exact(f,d) computes the exact probability.


# Define the functions sim() and exact().

sim<-function(f,d,nrep){
ans<-0
for(i in 1:nrep)ans<-ans+(0<min(tabulate(sample(d,f,replace=TRUE),d)))
ans/nrep
}

exact<-function(f,d){
if(d==1)return(1)
if(f<d)return(0)
i<-1:(d-1)
1-sum((-1)^(i-1)*choose(d,i)*((d-i)/d)^f)
}

# Problem 1.20 uses these values:
f<-12 # Number of friends
d<-7  # Number of days.

nrep<-1e6  # Number of simulation repetitions.

# Run three simulations.

sim(f,d,nrep)
sim(f,d,nrep)
sim(f,d,nrep)

# Compute exact answer.

exact(f,d)