Hint: One can write any real number $x$ as $x = \floor{x} + \{x\}$ , where $\{x\}$ is the fractional part of $x$ (for example, $3.41 = 3 + 0.41$ , and $0.41$ is the fractional part). Therefore, the condition is here is that $\{x+y\} = \{x\} + \{y\}$ . When does this hold true?

Hint: You can obtain a sum of $7$ by obtaining $(1,6), (2,5),$ or $(3,4)$ in some order on the dice. Add up the probabilities of obtaining some permutation of each of these pairs as a function of $x$ .

Hint: Claire bankrupts Jeff by either bankrupting him with their original stacks or by being bankrupt herself and then coming back to bankrupt him with her new stack. Consider these two cases separately.

Hint: Each order of breaking the targets is equivalent to some anagram of length $n_1 + \dots + n_r$ , where the $i$ th spot being $1 \leq j \leq n$ is saying that a target in column $j$ was shot on the $i$ th turn.

Hint: The best approach is to write $\var{X} = \ev{X^2} - (\ev{X})^2$ . Both of these can be computed via LOTUS.

Hint: Let $T$ be the amount of time that Dan takes to eat the cookie. We are looking to compute $\prob{T \leq 4 + 1 \mid T \geq 4}$ , as we want the probability that he finishes the cookie in the next minute given he took at least $4$ minutes.

Hint: The number of runs is equal to the number of starts of runs, as every single run must have a start.

Let $X_i$ be the indicator of the event that the $i$ th marble drawn out is the start of a new run. $T = X_1 + X_2 + \dots + X_{2n}$ gives the total number of runs.

All of these except one are identically distributed.

Hint: The radius of the circle is $A/2$ , so the area is $\frac{\pi}{4}A^2$ . The area of the ellipse is $\pi AB$ . Get an event in terms of $A$ and $B$ .

Hint: Sketch this region out in the plane and integrate over the joint PDF of $X$ and $Y$ .