# This function computes the probability that a poker hand contains
#  "nothing" (Exercise B5) by an exact method (a formula) and by a
#  simulation approximation.  

# The composition of the deck can be varied: nsuit is the number of
#  suits in the deck, nval is the number of card values, and nhand is
#  the number of cards in a hand.  The usual situation is nsuit=4,
#  nval=13, nhand=5.  In the simulation, the computer randomly deals
#  nrep hands of poker and determines what fraction of them contain
#  nothing.

pnothing<-function(nsuit,nval,nhand,nrep){
ndeck<-nsuit*nval
den<-choose(ndeck,nhand)
nseq<-pmax(nval-nhand+1,0)
PA<-choose(nval,nhand)*nsuit^nhand/den
PAB<-nseq*nsuit^nhand/den
PAC<-choose(nval,nhand)*nsuit/den
PABC<-nseq*nsuit/den
exact.ans<-PA-PAB-PAC+PABC
tot<-0
deck<-cbind(rep(1:nsuit,rep(nval,nsuit)),rep(1:nval,nsuit))
hm1<-nhand-1
for(i in 1:nrep){
hand<-deck[sample(ndeck,nhand),]
tot<-tot+((!anyDuplicated(hand[,2]))&&(diff(range(hand[,2]))!=hm1)&&(diff(range(hand[,1]))>0))
}
sim.approx<-tot/nrep
c(PA=PA,PAB=PAB,PAC=PAC,PABC=PABC,exact.ans=exact.ans,sim.approx=sim.approx)
}