Jamshrivelson

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]

Question: your friend Duane’s friend’s no-good, duplicitous, self-promoting grandchild has made extraordinary claims regarding their performance in the well-known cabin and road trip cardgame known as War. According to them, they’ve triumphed in a round where they won every single matchup with no ties, ending the game in a mere $26$ hands. Never one to let the feats of grandchildren go unchallenged, you set out to compute the probability of this occurrence. About how many rounds of War would one need to play to bear witness to their flawless victory? [Solution]

Question: Quality control at your ruler factory has taken a turn for the worst and your latest shipment of rulers are all broken into 4 pieces! What is the average length of the shard that contains the $\text{6 inch}$ mark? [Solution]

Question: It’s year 5 of quarantine and all computing power is reserved for face filters that make you look like a baby or the GEICO lizard. If you want to crunch numbers there’s a new calculator in town — the C. elegans petri dish. To add two numbers $x$ and $y$ you gather as many nematodes, put them in the dish, and come back the next day to see how many nematodes there are. The nematodes will pair bond (sex doesn’t matter, C. elegans are almost all hermaphroditic) and each pair will procreate (yielding one new worm), or not, with probability $1/2.$ It’s noisy, and it’s random, but that’s the best we can do in these trying times. To raise a number $x$ to the $n^\text{th}$ power, you gather as many nematodes and leave them in the dish for $n$ days, and however many there are upon your return is $x^n.$ Under these rules, what is $(1+1)^n$? [Solution]

Question: A gaggle of graduates gathers round a gargoyle who genuflects until they gear up in a circle upon which they (the gargoyle) announce, I thought no two of you were now in the same position relative to one another but alas I am mistaken, you comprise a number larger than $100$ such that you are the least populous class size for which it is simply not possible for less than $2$ of you to have remained in the correct relative position. How many students are in this graduation class? [Solution]

Question: Tortoise and Hare are having a stroll down a 10 mile road — Tortoise can run at a healthy clip of $60$ mph while Hare can run at an even healthier clip of $75$ mph. But that’s not all… the road is magic and every minute, on the minute, the road is stretched uniformly by $10$ miles. If Tortoise and Hare want to finish their stroll at the same time, how long should Hare hang back after Tortoise gets started? [Solution]