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1. Equivalence classes from given related pairs ●○○○○ 10 ptsLet $A=\{a,b,c,d,e\}$. Suppose $R$ is an equivalence relation on $A$. Suppose also that $aRd$, $bRc$, $eRa$ and $cRe$. How many equivalence classes does $R$ have?
2. Ace of spades before face cards ●●○○○ 20 ptsCarson flips over the cards of a standard 52-card deck one at a time. Find the probability that he flips over the ace of spades before any face card (jack, queen or king). Round to two decimal places.
3. Argument of a sum of exponentials ●●○○○ 20 ptsThe number $e^{7\pi i/60} + e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i/60} + e^{47\pi i/60}$ is written in the form $re^{i\theta}$, where $r \ge 0$ and $0 \le \theta < 2\pi$. Find $\theta$ in degrees.
4. Value of even plus odd function ●●○○○ 20 ptsLet $e(x)$ be an even function and let $o(x)$ be an odd function such that $e(x) + x^2 = o(x)$ for all $x$. Let $f(x) = e(x) + o(x)$. Find $f(2)$.
5. Value of f(3) from symmetry conditions ●●○○○ 20 ptsLet $f(x)$ be a function with domain $\mathbb{R}$ such that $f(x+1) = f(-x-1)$ and $f(2-x) = -f(x)$ for all real $x$. Find the value of $f(3)$.
6. Aces of spades and clubs adjacent ●●○○○ 20 ptsA standard 52-card deck is shuffled uniformly at random. Find the probability that the ace of spades and the ace of clubs are next to each other in the deck, in either order. Round to four decimal places.
7. Mean of a normal distribution ●●○○○ 20 ptsLet $X$ be normally distributed. Given that $P(-3<X<-1)=P(3<X<5)$, find the expectation $E(X)$.
8. First, fifteenth or last ball red ●●○○○ 20 ptsA box contains 100 balls, of which 40 are red. The balls are drawn from the box one at a time at random, without replacement. Determine: 1. the probability that the first ball drawn is red, 2. the probability that the fifteenth ball drawn is red, 3. the probability that the last ball drawn is red. Give each answer as a decimal.
9. Roots of a symmetric odd function ●●●○○ 50 ptsLet $y=f(x)$ be an odd function defined on $\mathbf{R}$ whose graph is symmetric about the line $x=1$. When $0<x\leq 1$, $f(x)=\log_3 x$. Find the sum of all real roots of the equation $f(x)=-\frac{1}{3}+f(0)$ in the interval $(0,10)$.
10. Rolling more 1s than 6s ●●●○○ 50 ptsIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1s than 6s? Round to two decimal places.

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