Question library

Roll $2$ fair $6$-sided dice independently. What is the probability that the total equals $9$?

Roll $3$ fair $6$-sided dice independently. What is the probability that the largest value shown is at most $4$?

Roll $2$ fair $6$-sided dice independently. What is the expected value of the larger of the $2$ results? Give an exact fraction.

A fair $6$-sided die is rolled repeatedly and independently, and the results are added up. You stop as soon as the running total is at least $7$. What is the expected number of rolls? Give an exact fraction in lowest terms.

You roll a fair $6$-sided die and are paid the number shown. After seeing any roll, you may reject it and roll again, but you may roll at most $3$ times in total, and the last roll must be kept if you reach it. Playing optimally to maximise the expected payoff, what is the expected payoff? Give an exact fraction in lowest terms.

A fair coin is tossed $6$ times, independently. What is the probability that all $6$ results are the same (all heads or all tails)? Give a fraction in lowest terms.

A fair coin is tossed repeatedly and independently. Pattern $\text{HTH}$ and pattern $\text{HHT}$ are both watched in the sequence of results ($H$ is heads, $T$ is tails), and overlaps with earlier tosses count. What is the probability that $\text{HTH}$ appears before $\text{HHT}$? Give a fraction in lowest terms.

A stock rises tomorrow with probability $0.4$. A model raises a bullish flag with probability $0.9$ when the stock will rise and with probability $0.2$ when it will not. The flag is raised. What is the probability that the stock rises?

$2$ fair $6$-sided dice are rolled independently. You are told that at least one of them shows a $6$. What is the probability that the sum is at least $10$?

An asset is worth $110$ or $90$ with equal probability. A trader arrives: with probability $0.25$ the trader is informed and buys if the value is $110$ and sells if it is $90$; otherwise the trader is uninformed and buys or sells with probability $\frac{1}{2}$ each. You, a market maker, observe a buy. What is the expected value of the asset given this buy?

A bag holds $3$ coins whose probabilities of heads are $\frac{1}{4}$, $\frac{1}{2}$ and $\frac{3}{4}$. One coin is chosen uniformly at random and flipped $3$ times independently, giving $3$ heads. What is the probability that the next flip of the same coin is heads?

A well-shuffled standard deck has $52$ cards, including $4$ aces. You draw $2$ cards without replacement. What is the probability that both cards are aces? Give a fraction in lowest terms.

A deck has $10$ cards numbered $1$ through $10$. You draw $2$ cards uniformly at random without replacement. What is the probability that the sum of the numbers is even? Give a fraction in lowest terms.

A standard deck has $52$ cards, of which $26$ are red. You deal $3$ cards uniformly at random without replacement. What is the probability that exactly $2$ of them are red? Give a fraction in lowest terms.

A deck has $20$ cards, and exactly $3$ of them are marked. The deck is shuffled uniformly and you flip cards one at a time until the first marked card appears. What is the expected number of cards flipped, counting the marked card itself? Give a fraction in lowest terms.

A biased coin lands heads with probability $\frac{3}{4}$ on each independent toss. It is tossed until the first tails appears. What is the expected number of tosses, counting the toss that shows tails?

A fair coin is tossed repeatedly with independent tosses. What is the expected number of tosses until $3$ heads in a row appear for the first time?

A fair coin is tossed $6$ times, with independent tosses. What is the probability that at least $3$ heads in a row appear somewhere in the sequence?

Desk A sends $60\%$ of all orders and desk B sends the other $40\%$. An order from desk A is faulty with probability $5\%$, and an order from desk B is faulty with probability $10\%$, independently across orders. One order is picked at random and it is faulty. What is the probability that it came from desk B? Give a fraction in lowest terms.

A box holds $5$ red chips and $3$ blue chips. You draw $2$ chips one after another without replacement, each remaining chip being equally likely. Given that the second chip is blue, what is the probability that the first chip is also blue? Give a fraction in lowest terms.

The market is in a high-volatility regime with probability $0.2$ and in a low-volatility regime with probability $0.8$. On each day, given the regime, a big move occurs independently with probability $0.7$ in the high regime and $0.1$ in the low regime. You observe big moves on $2$ consecutive days. What is the probability that the regime is high? Give a fraction in lowest terms.

A hand of $5$ cards is dealt from a well-shuffled standard deck of $52$ cards, which contains $4$ aces. Given that the hand contains at least $1$ ace, what is the probability that it contains at least $2$ aces? Give a fraction in lowest terms.

An urn holds $4$ green and $6$ white balls. You draw a ball at random, note its color and put it back, and you repeat this $3$ times in total. What is the probability that exactly $1$ of the drawn balls is green?

Urn A holds $3$ red and $2$ black balls. Urn B holds $1$ red and $4$ black balls. A fair coin picks one urn, and then a ball is drawn at random from that urn. The ball is red. What is the probability that it came from urn A? Give a fraction in lowest terms.

An urn holds $10$ tickets numbered $1$ to $10$. You draw $3$ tickets at random without replacement. What is the probability that the largest number drawn is exactly $7$? Give a fraction in lowest terms.

An urn holds $6$ red and $4$ blue balls. You draw balls one at a time at random without replacement and stop as soon as a red ball appears. What is the expected value of the number of draws, counting the final red draw? Give a fraction in lowest terms.

An urn holds $3$ red and $3$ blue balls. You draw all $6$ balls one at a time at random without replacement. What is the probability that, after every draw, the number of red balls drawn so far is at least the number of blue balls drawn so far? Give a fraction in lowest terms.

A walker starts at $0$ on the integer line and makes $4$ steps. Each step is $+1$ or $-1$ with probability $\frac{1}{2}$, independently. What is the probability that the walker is at $0$ after the $4$ steps?

A token sits at position $1$ among the positions $0$, $1$, $2$, $3$. Each move goes up by $1$ with probability $\frac{2}{3}$ and down by $1$ with probability $\frac{1}{3}$, independently. Positions $0$ and $3$ are absorbing. What is the probability that the token is absorbed at $3$?

A fair walker starts at position $2$ on positions $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$. Each step goes up $1$ or down $1$ with probability $\frac{1}{2}$, independently. Positions $0$ and $7$ are absorbing. What is the expected number of steps until absorption?