Solve the following quantitative reasoning problems. Show enough work to make each result auditable, state any interpretation you need, and keep exact fractions where practical. ### Clarifying Questions to Ask - Should sampled factory output be modeled as independent draws from a very large production stream? - For the round-table problem, may rotations be treated as equivalent? - Are flower start times continuous, so endpoint coincidences have probability zero? - For the canoe problem, is paddling speed constant relative to the water on both days? ### Part 1 - Animals in a Barn A barn contains only spiders, chickens, and cows. There are 520 legs in total. The number of chickens is twice the number of cows, and the number of spiders is twice the number of chickens. How many spiders are there? #### What This Part Should Cover - Converts the population ratios into one linear equation and checks integrality. ### Part 2 - Circular Seating Seat Anna, Brian, Charlie, Dixie, and Eva around a round table. Anna cannot sit next to Brian or Eva. Brian cannot sit next to Charlie. Dixie cannot sit next to Eva or Charlie. If Dixie sits immediately to Anna's left, who sits immediately to Brian's left? Use one consistent orientation for “left.” #### What This Part Should Cover - Uses adjacency constraints and explains why the remaining arrangement is forced up to rotation. ### Part 3 - Conditional Survey Probability The exercise survey below records 1,000 people by age group: | Exercise | 18-22 | 23-27 | 28-32 | 33-37 | Total | | --- | ---: | ---: | ---: | ---: | ---: | | Run | 54 | 40 | 42 | 66 | 202 | | Bike | 77 | 68 | 90 | 70 | 305 | | Swim | 28 | 43 | 50 | 52 | 173 | | Other | 90 | 78 | 71 | 81 | 320 | | Total | 249 | 229 | 253 | 269 | 1000 | Given that a surveyed person is 33 years old, what is the probability that the person prefers swimming? #### What This Part Should Cover - Conditions on the correct age-group column rather than the full survey. ### Part 4 - News and Likes Thirty percent of stories are fake. A fake story has probability 0.80 of receiving at least 100 likes; a real story has probability 0.08. Given at least 100 likes, what is the probability that the story is fake? #### What This Part Should Cover - Applies Bayes' rule with both fake and real paths in the denominator. ### Part 5 - Factory Identification Factory A produces 40% red widgets and Factory B produces 80% red widgets. Select a factory uniformly, then independently sample two widgets from its large output. Both are red. What is the probability that Factory A was selected? #### What This Part Should Cover - Updates equal prior factory probabilities using the likelihood of two red draws. ### Part 6 - Three-Dice Game Roll three fair six-sided dice. Three equal values earn $20, exactly one matching pair earns $10, and three distinct values lose $2. What is the expected return per roll, rounded to the nearest cent? #### What This Part Should Cover - Counts all three mutually exclusive outcome classes and forms the expectation. ### Part 7 - Higher of Two Dice Roll a fair six-sided die twice. If the values differ, the payoff is their maximum in dollars; if they match, the payoff is zero. What is the expected payoff? #### What This Part Should Cover - Accounts for ordered unequal pairs and excludes equal rolls from the maximum payoff. ### Part 8 - Muffin Inventory Five customers independently buy one item each. Each chooses a muffin with probability 0.30 and a croissant otherwise. Only two muffins remain. What is the probability that no customer who wants a muffin finds the muffins sold out? #### What This Part Should Cover - Recognizes that muffin demand must be at most two and evaluates the binomial tail. ### Part 9 - Truthful Cat Owners Asta, Bronya, and Clara own distinct positive integer numbers of cats. A speaker always tells the truth when addressing someone with fewer cats and always lies when addressing someone with more cats. 1. Bronya tells Clara, “You have the most cats.” 2. Asta tells Bronya, “I have exactly 30% more cats than you.” 3. Asta tells Clara, “Your count is the average of mine and Bronya's.” 4. Clara tells Asta, “You have at least four more cats than I do.” How many cats does Asta own? #### What This Part Should Cover - Infers the ordering from truth conditions, enforces integer counts, and uses the final inequality. ### Part 10 - Canoe Trip Two friends paddle upstream for four hours, realize their campsite is downstream, then paddle downstream for five hours to reach it. The next day they paddle 23 miles upstream from the campsite to their original starting point, arriving at 16:00. The river flows at 2 mph and their speed relative to the water is constant. When did they leave the campsite? Give `hh:mm`. #### What This Part Should Cover - Derives paddling speed from the first day's net displacement, then computes upstream travel time. ### Part 11 - Cookie Arrangements There are five indistinguishable snickerdoodles and seven indistinguishable chocolate-chip cookies. How many length-six sequences can be made without exceeding either inventory? #### What This Part Should Cover - Sums binary sequences by the feasible count of snickerdoodles. ### Part 12 - Choosing a Losing Cat Three cats win a contest with probabilities `3/5`, `3/10`, and `1/10`. You choose one uniformly to support and learn that it did not win. What is the probability that you chose the cat whose win probability is `3/5`? #### What This Part Should Cover - Conditions on losing and includes the different loss probabilities of all three choices. ### Part 13 - Spinner Distinct Regions A spinner lands in three regions with probabilities `1/6`, `1/6`, and `2/3`. What is the expected number of spins needed to have landed in two distinct regions? #### What This Part Should Cover - Conditions on the first region and uses a geometric waiting time for a different region. ### Part 14 - Overlapping Blooms Purple and red flowers independently choose start times uniformly over the next 30 days. Purple blooms for 9 days and red for 12 days. What is the probability that their bloom intervals overlap at some time? #### What This Part Should Cover - Represents start times in a square and subtracts the two non-overlap triangles. ### What a Strong Answer Covers - Shows a reproducible derivation for every part rather than listing guesses. - Uses conditional probability, counting, expectation, and algebra with the correct sample spaces. - States interpretations that affect an answer and checks the result against basic bounds. ### Follow-up Questions 1. How would the factory answer change with a non-uniform prior over factories? 2. Can you derive a general formula for the two-dice payoff with an `m`-sided die? 3. How would the flower-overlap probability change for a finite horizon with arbitrary bloom durations?