SIG interview questions

A barnyard fair hosts geese, goats, and spiders. A careful tally turns up $440$ legs in total. There are twice as many geese as goats, and twice as many spiders as geese. How many spiders are at the fair? (Geese walk on $2$ legs, goats on $4$, spiders on $8$.)

A boardwalk arcade stall runs a game built around a single fair six-sided die. Two wagers are on offer. Wager 1: put down $4 on the roll landing even; a winning ticket pays $6 back in total. Wager 2: put down $6 on the roll landing odd; a win returns $9 in total. You can buy as many tickets of either type as you like. If your goal is to maximize expected profit, what is the right amount to wager?

Shares of Helios Robotics will do exactly one of two things by tomorrow's open: double, or fall by half. Separately, a colleague is happy to take either side of an even-money (1:1) wager on which direction the stock moves. Suppose you put $A$ whole dollars into the stock and stake $B$ whole dollars on the wager, choosing sides so that your total profit comes out identical whether the stock jumps or drops. What are the smallest whole-dollar values of $A$ and $B$ that lock in a positive profit either way?

A jeweler has $8$ gold rings that look identical, but exactly one contains extra metal and is a bit heavier than the rest. Using only a two-pan balance (no weights), how few weighings guarantee finding the heavy ring?

In a home poker game the pot has grown to $\$80$ when your last remaining opponent bets $\$40$ into it. Reading the board and her range, you put your chance of winning the hand at $30\%$ if you call and see it through, with no further betting after the call. Is calling the $\$40$ profitable? Show the expected value.