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 9 Drawing two cards from a deck of 16 (4 ranks and 4 suits) 5 For each pair of events A and B, find P(A|B) and P(B|A). 3 Probability Independence - Determining if two sets are independent (drawing two cards) 3 Prove: if $a ∈ [x]$ and $b ∈ [y]$, then $a + b = x + y$ (mod $n$). 2 Let $G$ be a planar graph. Suppose that $V = V_4 \cup V_8$. Prove that if $|V_8| = 12$, then $|V_4| \ge 18$

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 2 Pigeonhole Principle - If I play a hand of Texas Hold 'Em per minute for a day, prove I will be dealt a particular pair of cards at least twice. 1 Suppose $P (1) + P (2) = 3P (3)$ and $P (2) + P (3) = P (1)$ for a sample space $(S,P)$ . Find the probability function. 1 Find a crossing-free planar embedding of this graph? 1 Given finite sets $A$ and $B$, let $I = \{f: A \rightarrow B : f$ is one to one$\}$ and $S = \{f : A \rightarrow B : f$ is onto$\}$. Compute |I||S|.

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