#1. Describe why the formulas below count the specified hands. (A sketch of a  suitably labeled tree will usually be sufficient.  Though the below version of how to compute a "Pair" will require at least one sentence on your part to describe what is going on.) 

#2. Compute the probability of being dealt each of the below hands.  

#3. Suppose you "flip a coin" by dealing out a poker hand and calling it Heads if you have anything other than just a "High Card" and Tails otherwise.  Is this a fair coin?  How close to being fair is it?

#4. How many times more likely is a Streak than a Flush?   Some people claim this is unintuitive. Give a possible reason why this might feel unintuitive to an experienced poker player.  

#5.  (Extra Credit) Be the first one to find an error in the below list and get a free HW assignment!  (In other words, a score of your choice is turned into a maximal score. I give this EC since I foolishly forgot to save our poker day's R-session, and was forced to recompute the below list rather quickly.  It is very easy to make a silly error in making these counts, and I probably did.  Find it!)

############ For us a poker hand is simply 5 cards dealt out from a randomly mixed up standard 52 card deck (all hands are viewed as equally likely).  Recall, the different types of poker hands described below are mutually exclusive (for example four of a kind does not also count as a three of a kind).  In particular, this  list will add up to "Total". 
 
# Total Number of Hands
Total<-choose(52,5)

# Royal Flush
RF<-4

# Straight Flush
SF<-10*4-RF

# Four of a Hind 
FOK<-13*48

# Full House
FH<-13*12*choose(4,3)*choose(4,2)

# Flush
Fl<-4*choose(13,5)-SF-RF

# Streak
St<-10*4^5-SF-RF

# Three of a Kind
TOK<-13*choose(4,3)*choose(48,2)-FH

# Two Pairs
TP<-choose(13,2)*choose(4,2)*choose(4,2)*(52-8)

# Pair 
Pr<-13*choose(4,2)*choose(48,3)-2*TP-FH

# High Card
HC<-Total-(RF+SF+FOK+FH+Fl+St+TOK+TP+Pr)

# All hands in order
All<-c(RF,SF,FOK,FH,Fl,St,TOK,TP,Pr,HC)

All