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{\Large
MATH 2113/CSCI 2113, Discrete Structures II

Winter 2008

Toby Kenney

\large Homework Sheet 3

Due: Wednesday 30th January: 1:30 PM}
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\subsection*{Compulsory questions}

\item[1] 
(a) What is the probability that a randomly dealt 5-card poker hand is
  a straight? [But not a straight-flush.]

(b) What is the probability that a randomly chosen 5-card poker
  hand contains exactly one king?

\item[2] How many numbers between 100 and 1000 inclusive are
  divisible by at least one of 2, 3 or 5?

\item[3] Suppose we have 3 dice: one red, one blue, and one green, and
  we roll all 3, what is the probability that the number on the green
  die is at least as big as either of the other numbers, and the
  number on the red die is no bigger than either of the other numbers? i.e. 
${rm red}\leqslant{\rm blue}\leqslant{\rm green}$.

\item[4] A fair coin is tossed 5 times.

(a) What is the probability that the sequence HHH occurs (consecutively) 
somewhere in the 5 tosses?

(b) What is the probability that the sequence THT occurs (consecutively) 
somewhere in the 5 tosses?

\item[5] $n$ fair dice are rolled. 

(a) What is the probability that the highest number shown is a 5?

(b) For which value of $n$ is this probability greatest? [Hint: compare probabilities 
for consecutive values of $n$, to see if they are increasing or decreasing.]

\item[6] In the game Craps, played in casinos, the player first rolls 2 dice 
and adds them. If the total is 7 or 11, the player wins. If the total is 2, 3 or 
12, the player loses. If the total is anything else (4, 5, 6, 8, 9, or 10) 
this total is recorded and called the point. Now the player 
continues to roll two dice until the total is either 7 or is equal 
to the point. If it is 7, the player loses. If it is the point, the 
player wins.

(a) Suppose that after the first roll, the point is 5. What is the probability that 
the player goes on to win?

\subsection*{Bonus question}

(b) What is the probability that the player wins overall?

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