Jamshrivelson

Question: it’s your last chance to be on Jeopardy and you want to leave it all on the field. You’ve trained for years and are confident that you can answer every question correctly and beat your competitors to the buzzer every time. Your only issue is how to order your guesses. The Daily Double tile is placed randomly on the board, so you can’t predict its location. The two strategies you’ve narrowed it down to are guessing all the tiles in succession (all $\$200$ tiles then all $\$400$ tiles and so on) or to guess tiles randomly. What is the expected value of each strategy, assuming you have the whole “getting the answers right” part covered? [Solution]

Question: after a life of flipping fair coins, you’ve had enough. Your new mission is to construct elaborate and worthless schemes to extract virtual fair coins out of $3$ flips of a crooked one. Ever the positivist, you want to know just how many crooked coins there can be that suit your campaign. For how many values of $p$ can a fair coin be simulated using no more than $3$ flips? [Solution]

Question: you’re practicing for the forgotten NYC marathon. On the first day you set your treadmill to $9$ minutes per mile. On the second day, you set it to steadily “speed up” over time from $r_0 = 10$ minutes per mile at the beginning down to $r_1 = 8$ minutes per mile at the end (when the run is half over, the treadmill moves at $9$ minutes per mile). Which run took less time? [Solution]

Question: It’s Halloween Eve and famous ambassador (ambassador of pumpkins) David S. Pumpkins has appeared out of the clear blue sky to invade your apartment complex’s elevator. “Join me”, he entices, for a few rounds of “Damn That’s a Hot Pumpkin” his crazy new game that’s been sweeping the nation. Altogether, there are $61$ people, including David S. Pumpkins himself. The rules are simple, everyone gets in a circle, including David S. Pumpkins, and starts counting from $1$ to a number $N$ that’s specified by David S. Pumpkins himself. When his watch strikes “pumpkin time” the players start passing the pumpkin to the left, counting the numbers one by one, until they get to $N.$ When this happens, the person who said $N$ is eliminated, and the next round begins, starting with the person to their left. Distrusting of his pumpkin ways, you hang back and watch a few rounds from the wall. You see that the first person to be eliminated was $18$ positions to the left of David S. Pumpkins, the second person was $31$ positions to the left of whoever started their round, and the third person eliminated was the very person who began round three! What is the smallest value of $N$ for which this would be possible? After solving that, David S. Pumpkins proffers another: who will win this game? After solving that, David S. Pumpkins has had it up to here with other people winning his hot pumpkin games, and says “I’ll show you how to do it!!1” Putting aside the constraints from the first two problems, what value of $N$ should David S. Pumpkins pick so that he himself is the person to win the game Damn That’s a Hot Pumpkin? [Solution]

Question: It’s the day after the championship parade and you are Lebron James. Doc Rivers has been deposed, Paul George’s name is on the trade winds, and Patrick Beverley hasn’t thought about trash talk in over a month. Needless to say, life is good. But one day, your brother in arms, The Brow, Anthony Davis himself, challenges you to some $1$ on $1,$ first to score wins! Due to his dominant height and wingspan, he’s able to grab every single rebound, even the ones off his own three pointers! “What’s a king to do?”, you wonder. But then you realize — your career steal line is a full $0.2$ steals per game higher than Davis’. Every possession, you have the chance to steal the ball away with probability $L_\text{steal}.$ If yours and Anthony’s shot percentages are $L_\text{score}$ and $D_\text{score},$ respectively, then what is the chance that you, Lebron James, win the $1$ on $1$ game? Inversely, how high does your steal probability, $L_\text{steal},$ need to be for it to be an even match? [Solution]

Question: Like a banshee on a motorcycle, November approaches and brings with it everyone’s favorite day: the election… But as we know, nobody can conduct a poll these days, what with phones and computers and all that. That means you have no idea what’s going to happen. In fact, you might as well pull a random election result out of a hat. The only way the nightmare can end is if one candidate manages $50\%$ of the vote or more, or else it goes to a runoff. If there are $N$ candidates running in your town, what is the probability that a runoff will be necessary? [Solution]

Question: The ghost of Bob Parker takes a break from saving animals to remind you of your invitation to pandemic Price is Right, with Drew Carey. Usually, you’re up against some real price wizards, but since nobody has been to a store in $7$ months, everyone is on a level playing field and has no idea about prices. With the prices effectively random, you’re left to pure strategy to prevail at the game. As a refresher, everyone guesses the price of the product in succession such that the later contestants can hear the guess of the earlier contestants. Whoever guesses the highest without exceeding the true price of the item wins. In the case that everyone guesses too high, the game goes to whoever guessed closest. If you are the first to guess, and all the players are perfectly rational, what is the probability that you win the game? [Solution]

Question: You’re driving your truck — which has the footprint of a bicycle — down to the grocery store to pick up yams for the big day. Boy do you love yourself some yams! When you’re halfway there, you realize you left your N-95 face mask at home. What a klutz! You make a U-turn, pointing holding your front tire $30\,^\circ$ to the left until you turn around. When you’re almost home, you stroke your chin, contemplating your own forgetfulness, when you realize the mask has been on your face the whole time! Always one to outdo yourself, you decide that for your next U-turn, you’re going to angle the back tire a full $30\,^\circ$ of its own! What are your turning radii in either case? [Solution]

Question: Every night your children change their favorite word and refuse to let you go to bed until you guess which word it is. If you guess wrong, they’ll tell you if their word comes before or after yours in a list of all words. If there are $W = 267,751$ words that they’re knowledgable about, how many guesses will you take, on average, to uncover their word? [Solution]

Question: You’ve entered the annual Tour de 538 but, like a fool, you’ve gotten into a blowout fight with your teammate just before the race. As a result, they refuse to cooperate with you, claiming “the team is dead.” Not a day has passed, but you rue it already. The other two people in the race still have their team spirit intact, and they’ll cooperate, allowing one another to draft up so that they always finish one after the other. If, drafting aside, all cyclists are equally skilled, and the point allotment goes $\{5,3,2,1\},$ what is your expected point total? [Solution]