Three points are chosen uniformly at random on a circle. What is the probability that the triangle they form is acute?

Full solution and 2,800+ more at quantvault.org/problems?id=21

A box contains 4 blue balls and 3 red balls. You draw balls one at a time without replacement. Each blue ball earns you $1 and each red ball costs you $1. After each draw you see the ball's color, collect or pay the dollar, and then decide: stop and keep your running total, or draw again.

You may stop at any point, including before drawing at all, locking in $0. What is the optimal stopping strategy, and what is the expected profit under that strategy?

Full solution and 2,800+ more at quantvault.org/problems?id=122

You have an opportunity each day to trade an ETF mean-reversion signal. Each day the trade either wins or loses:

- With probability p = 0.54, you earn +1% on the capital you deploy.

- With probability q = 0.46, you lose -1% on the capital you deploy.

You choose a fraction f in [0, 1] of your total bankroll to invest in the trade each day. The remaining fraction 1 - f sits in cash and earns nothing.

1. Derive the fraction f* that maximizes the expected log-growth of your bankroll over one day, E[log(W1 / W0)].

2. What is the maximum expected log-growth at f*?

Full solution and 2,800+ more at quantvault.org/problems?id=117

There are n passengers boarding a plane with assigned seats 1, 2, ..., n. Passenger 1 has lost their ticket and picks a seat uniformly at random from all n seats. Each subsequent passenger k (for k = 2, 3, ..., n) takes their assigned seat if it is available; otherwise, they pick uniformly at random from the remaining empty seats.

1. Compute the probability that passenger n sits in their own seat (seat n).

2. Compute E[number of passengers who do NOT sit in their assigned seat].

Full solution and 2,800+ more at quantvault.org/problems?id=22

Alice and Bob are watching a single stream of fair coin flips.

Alice wins as soon as the pattern HTH appears. Bob wins as soon as the pattern HTT appears. Whichever pattern shows up first in the sequence determines the winner.

What is the probability that Alice wins?

Hint: both patterns start with the same two-letter prefix. Once the stream reaches HT, the next flip decides the race.

Full solution and 2,800+ more at quantvault.org/problems?id=501

A fair coin is tossed repeatedly. The game ends the moment either the pattern HH or the pattern HTH appears as consecutive flips, whichever comes first.

What is the probability that HTH appears before HH?

Model the process as a Markov chain over states that track the relevant suffix of the flip sequence. Set up and solve the system of linear equations exactly.

Full solution and 2,800+ more at quantvault.org/problems?id=104

You want to design a 6-sided die whose faces take values from {1, 2, 3, 4, 5, 6}, with repeats allowed. The requirement is that when you roll two identical copies of this die, every integer sum from 2 through 12 must be achievable with positive probability.

Two dice are considered the same if they have the same multiset of face values, meaning the same number of faces showing each value, regardless of physical orientation. Each of the 6 faces is equally likely on each roll.

How many distinct dice satisfy this property?

Full solution and 2,800+ more at quantvault.org/problems?id=78

You are offered the following game: roll a fair six-sided die up to 3 times. After each roll, you decide -- stop and collect the face value in dollars, or roll again and forfeit what you just saw. If you roll a third time, you must accept whatever comes up.

What is the optimal stopping strategy, and what is the expected payout under that strategy?

Full solution and 2,800+ more at quantvault.org/problems?id=14

Let B_t be a standard Brownian motion starting at 0. Fix a, b > 0 and define the first exit time tau = inf {t >= 0: B_t is not in (-b, a)}, so tau is the first time the process leaves the interval (-b, a).

1. Compute P(B_tau = a), the probability that the process exits through the upper barrier.

2. Compute E[tau], the expected time to exit.

3. Interpret part (1) as the fair odds for a one-touch wager with asymmetric barriers. If a bookmaker offers a bet that pays $1 when the process first hits a, and nothing if it hits -b first, what is the fair price?

Full solution and 2,800+ more at quantvault.org/problems?id=71

You have 12 balls that look identical. One of them is the odd ball - it is either heavier or lighter than the rest, but you do not know which. You have a two-pan balance scale.

1. Design a strategy that guarantees you can identify the odd ball AND determine whether it is heavier or lighter, using the minimum number of weighings.

2. Prove that your strategy is optimal by establishing an information-theoretic lower bound on the number of weighings required.

Full solution and 2,800+ more at quantvault.org/problems?id=46