The questions are similar to the ones you can find online, but there are some variants. All the specific numbers and shapes are randomly generated. Below is a summary done with AI, in shuffled order.
1. Probability and Bayes' Theorem
Factory A produces $a%$ red parts and $(100-a)%$ black parts, and Factory B produces $b%$ red parts and $(100-b)%$ black parts. A factory is picked at random and $k$ parts are drawn, all red. Find the conditional probability that these $k$ parts came from Factory A.
Three cats (or contestants) race, with winning probabilities $x$, $y$, $z$. You pick a cat at random to cheer for. Given that it did not win (or did win), find the probability that you picked the strongest (or weakest) cat.
A customer buys item A with probability $p$ and item B with probability $1-p$. The store has only $M$ of one item left, and there are $N$ people in line. Find the probability that those $M$ items are exactly enough / enough to satisfy everyone who wants that item.
Two kinds of flowers each start blooming on a random day within the next $D$ days, with bloom periods of $d_1$ days and $d_2$ days. Compute the probability that both are in bloom at the same time at some point.
2. Expected Value and Game Strategy
Roll $3$ fair $N$-sided dice. If all three numbers are the same you win $$A$; if exactly two are the same you win $$B$; if all are different you win/lose $$C$. Compute the expected payoff of a single roll.
A deck contains $M$ ranks with $N$ cards of each rank. Deal $K$ cards at random (without replacement) and compute the expected number of pairs in the hand.
You start with $A$ chips and your goal is $B$ chips. Each round you use Bold Betting (bet the smaller of the chips needed to reach the goal and everything you have), with win probability $p$. Find the probability of reaching $B$ chips before going broke.
A spinner has $N$ regions, with probabilities $p_1, p_2, \dots, p_N$ of landing in each. Compute the expected number of spins needed for the spinner to land in $K$ different regions for the first time.
3. Combinatorics and Grid Counting
A frog moves from $(0,0)$ to $(X,Y)$, and each step can only go up or right by 1 unit, and it cannot go in the same direction for $k$ steps in a row or more. Find the total number of valid paths.
There are $A$ identical items of type 1 and $B$ identical items of type 2. Choose $C$ of them and line them up in a row. Find the number of distinct arrangements.
Fill an $N \times N$ grid with the numbers $1 \sim N$ so that no number repeats in any row or column. Given the "visible buildings" constraints around the edges, solve for the values of the specified cells.
4. Logic Puzzles
Several people hold different numbers of items. The speaking rule is "tell the truth/lie to people who have fewer items than yourself, lie/tell the truth to people who have more". From the statements, work out the number of items a particular person has.
$N$ people sit around a round table for dinner. From geometric and positional constraints such as "A is not next to B/C" and "D sits to the right of A", determine a particular person's relative position.
A multi-level lever hanging system is balanced, and the total weight of the system is known to be $W$ pounds. From the lever arm lengths and the balance relations at the nodes, solve for the weight of a particular shape (for example the green triangle).
Given the downstream rowing time $t_1$, upstream rowing time $t_2$, the current speed $v$ and the total displacement, work out the exact time of day the trip started, using the final arrival time.
Given a profit cross-table over multiple years and multiple departments, and under a stated condition (such as "departments/years with profit greater than $$X$"), compute the conditional probability that a random sample belongs to a particular year or department.
There are also some random questions, like a combinatorics one about counting cookie combinations.
The output format is stated clearly. Sometimes it is an integer and sometimes you keep two decimal places.
Discussion
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