2026 Jane Street Interview Questions

Quant Interview Questions

90 Questions

671

You pick n points independently and uniformly at random on the circumference of a circle. 1. What is the probability P(n) that all n points lie within some semicircle of the circle? 2. Simplify your answer to a closed form in terms of n.

Quantitative Researcher Interview

670

How much would you pay to play a game where you roll a die and receive the number of dots rolled in dollars?

Analyst Interview

668

What is the probability of drawing two red cards in succession from a standard deck of 52 cards and from a double deck of 104 cards?

Analyst Interview

669

An unfair die with 12 faces has the number 11 with a probability of 40%, while the other faces are equally likely. You and another player are playing a game where whoever is closer to the correct answer wins. What is your optimal strategy, and should you choose to go first or second?

Analyst Interview

667

What is the smallest positive integer whose digits multiply to 108?

Analyst Interview

666

An escalator travels upwards at X steps per second. You walk up the escalator at Y steps per second, and the escalator is Z steps tall. How long will it take you to reach the top?

Quant Intern Interview

665

What is the sum of the even numbers from 1 to 60?

Quant Intern Interview

664

What is the expected number of rolls of a six-sided die to observe each face at least once? What is the expected number of rolls to observe each face at least twice?

Quant Intern Interview

663

You have 3 buckets, each able to hold one ball. Every time you throw a ball, you are guaranteed to hit a bucket; if the bucket is already filled, the ball bounces out. Each ball that goes in gives +1, each ball that bounces out gives -1. If you have 2 balls to throw, what is the expected gain?

Quant Intern Interview

662

Player 1 chooses a number, then Player 2 chooses a different number. Two dice are tossed, and whoever is closer to the outcome (the sum of the two dice) wins. Would you rather be Player 1 or Player 2? What number would you choose?

Quantitative Trading Interview

660

You flip a coin 4 times. What is the probability of getting exactly 2 heads?

Quantitative Trading Interview

661

I am thinking of a number that does not contain the digit 1. The product of its digits is 96. What is the smallest possible number? What is the largest?

Quantitative Trading Interview

659

It rains 30% of the time on Saturday and 40% of the time on Sunday. How often does it rain on the weekend?

Quantitative Trading Interview

658

If there is a 0.3 probability it will rain on Saturday and a 0.4 probability it will rain on Sunday, what is the range of possible probabilities that it will rain on at least one day?

Quantitative Trading Interview

657

Devise a strategy for playing a game where a fair 100-sided die is rolled, and you are paid the value shown on the die in pounds. Each roll of the die costs £1.

Quantitative Trading Interview

656

Find the smallest integer such that the product of its digits is 3,000,000.

Quantitative Trading Interview

655

What is the minimum number of people required to guarantee that there exist seven people who are born in the same month?

Quantitative Trading Interview

654

You roll a 6-sided die and a 10-sided die. I will pay you the sum of the numbers on the dice only if you correctly guess the sum before the roll. What number should you guess to maximize your expected payout?

Quantitative Trading Interview

643

You have a drawer with an infinite supply of socks in a 1:1 ratio of two colors. What is the expected number of draws needed to get a matching pair?

Quant Trading Intern Interview

653

There are 30 people, consisting of 15 couples. Everyone shakes hands with other people, except nobody shakes hands with their own partner. You are told that 29 people each shook hands with a different number of people (for example, one person shook hands with 4 people, another with 5, etc.). How many people did the 30th person shake hands with?

Quant Trader Interview

652

Two teams, A and B, played football twice in independent matches. In match 1, the average number of shots by team A is greater than that of team B. In match 2, the average number of shots by team A is also greater than that of team B. Can we conclude that, when combining data from both matches, the average number of shots by team A is still greater than that of team B? If so, why? If not, provide an example.

Quant Trader Interview

651

How many people do you need to have in a room before you can be certain that at least 5 of them were born in the same month?

Quant Trader Interview

650

How can you simulate the roll of a fair 15-sided die using a standard 6-sided die?

Quant Trader Interview

649

What is the weight of each of five melons if their pairwise weights sum to 16, 17, 18, 19, 20, 21, 22, 24, 25, and 26?

Quant Trader Interview

648

If you flip a fair coin, what is the probability of getting a head? What is the probability of getting a certain number of heads if you flip two coins? How does this generalize for n coins?

Quant Trader Interview

647

You are playing a one-player game with two opaque boxes. At each turn, you can choose to either 'place' or 'take.' 'Place' puts $1 from a third party into one box randomly, while 'take' empties out one box randomly and the money is yours. The game consists of 100 turns, and at each turn you must either place or take. Assuming optimal play, what is the expected payoff of this game? Note: You do not know how much money you have taken until the end of the game.

Quant Trader Interview

646

What is the probability of getting an even number of heads when tossing 4 coins?

Quant Trader Interview

645

1. One bus leaves. At each stop, three-quarters of the people get off the bus, while seven get on. It continues this way until the end of the route. What is the minimum number of passengers at the beginning? 2. Roll two dice and multiply their results. What is the probability that the product is a perfect square?

Intern Quant Trading Interview

644

For an odd number of fair coin flips, what is the probability of getting an even number of heads? What about for an even number of flips? How does this probability change if some of the coins are unfair?

Quant Trading Intern Interview

642

You play a 100-round game: in each round, you can either roll a fair 20-sided die (with outcomes 1 to 20) or 'lock in' the current face value and collect that same reward for all remaining rounds. You must roll in the first round. What is your optimal stopping or locking strategy, and what is your expected value?

Quant Trading Intern Interview

640

Starting with 13 red and blue cards labeled from 1 to 13, remove 7 blue cards at random. What is the probability that a card drawn is blue, given that its number is 3?

Summer Intern Interview

641

I have a 12-sided die and you have a 20-sided die. Each of us gets up to two rolls, and on either roll, we can choose to stop and keep the number from that roll. Whoever has the higher number wins, with ties going to the person with the 12-sided die. What is the probability that the person with the 20-sided die wins this game?

Summer Intern Interview

638

If the chance of rain on Saturday is 70% and on Sunday is 60%, what is the probability that it will rain at least once during the weekend?

Junior Trader Interview

639

You have 3,000 apples to transport from city A to city B, which are 1,000 miles apart. When there are apples on the truck, one apple is consumed per mile driven. Assume unlimited gasoline. What is the maximum number of apples that can be transported to city B? Given your answer is x, what market would you make on a lottery: you win $10 if your answer is correct, otherwise $0. (All calculations must be done without pen and paper.)

Junior Trader Interview

637

Imagine an infinite chessboard. If a knight starts from a particular square, in how many different squares can it possibly end up after 10 moves? You do not need to calculate an exact number, but instead provide a 95% confidence interval for the number of possible ending squares. Pen and paper allowed.

Junior Trader Interview

636

I have a number in mind such that the product of each of its digits is 96, and no digit is 1. What are the largest and smallest numbers I could think of?

Junior Trader Interview

634

What is the cube root of 169381?

Junior Trader Interview

635

Choose a subset of numbers from 1 to 30 such that no two elements in the subset share a common factor (i.e., they are pairwise coprime). What is the subset with the largest possible sum?

Junior Trader Interview

632

What is the probability of getting an even number of heads in seven coin tosses?

Junior Trader Interview

633

A train is approaching a bridge, and you are standing at the 1/4 position along the length of the bridge. If running in either direction allows you to escape just in time, what is the ratio of your speed to the speed of the train?

Junior Trader Interview

631

Two fair 6-sided dice are rolled. What is the probability that the product of the top numbers is a perfect square?

Junior Trader Interview

630

You flip a fair coin an unlimited number of times. If the sequence HHT appears before HTT, you win. What is the probability that you win?

Junior Trader Interview

628

What is the smallest number made up of only 1s and 0s that is divisible by 225?

Trader Intern Interview

629

What is the expected value of a die roll? Additionally, create a market (bid-ask spread) based on your expectation.

Trader Intern Interview

627

In Russian Roulette, there is a revolver with 4 blanks and 2 bullets loaded consecutively in adjacent chambers. If the person before you fires a blank (the gun did not discharge), should you take the next shot as-is, or spin the cylinder before shooting? Justify your answer with probabilities.

Trader Intern Interview

626

You are playing a game in which four fair coins are flipped, and the amount of money you receive in dollars equals the number of heads that appear. If you do not like the outcome of the first four flips, you may re-flip all four coins, but you must accept the result of the second set. Determine the fair value for the game.

Trader Intern Interview

625

You have 4 coins. You throw them in the air. For every coin that lands heads, you get 1 dollar. How much would you pay to play this game?

Trader Intern Interview

624

What is the expected outcome when rolling two 10-sided dice? Please explain why.

Trader Intern Interview

623

What is 1,000 to the 1,000th power?

Trader Intern Interview

622

What is the minimum number of people required in a group to ensure that at least 7 people share the same birthday month?

Trader Intern Interview

618

You have two bowling balls of the same density. One has a radius of 8 and a weight of 16; the other has a radius of 12. What is the weight of the second ball?

Trader Intern Interview

621

Two players toss a fair coin repeatedly. Player A wins if the sequence HHT appears first; player B wins if HTT appears first. What is the probability that player A wins the game?

Trader Intern Interview

620

Two people each bid a number before rolling a 30-sided die. Whoever bids closer to the number the die shows wins, and wins an amount of money equal to the number rolled. For example, if I bid 15 and you bid 16, and the die lands on 10, then I win 10 from you. What is the optimal bidding strategy and the expected payoff?

Trader Intern Interview

619

What is the expected value of an optional reroll of a fair six-sided die, given that your payout is the number of pips shown on the die? In other words, how much would you pay for the option to reroll once, given that your final payout is the number shown on the die?

Trader Intern Interview

617

Basketball players A and B each play in Game 1 and Game 2. In both games, A has a higher shooting average than B. Is it possible for B to have a higher overall shooting average than A? If so, provide an example.

Trader Intern Interview

616

A and B play a game. Each chooses a different integer between 2 and 12. Two dice are rolled, and the sum is calculated. The player whose chosen number is closer to the dice sum wins. If you can choose first or second, which position should you choose, and what is your optimal strategy?

Trader Intern Interview

615

What is the set of numbers between 2 and 30, where no two numbers share a common factor greater than 1 (i.e., the set is pairwise coprime), that gives the maximum possible sum? Using the same rules, what is the highest possible number you can have in a set of 1000?

Trader Intern Interview

614

If you randomly pick a 3-digit number, what is the probability that all three digits are even numbers?

Trader Intern Interview

613

Walk me through the solution to the birthday problem (i.e., calculate the probability that at least two people in a group share the same birthday).

Trader Intern Interview

612

How much would you pay for a game where your payoff equals the number shown on a die, with an option to reroll once? Generalize to n opportunities to reroll.

Trader Intern Interview

611

What is the probability of getting at least one head in 4 coin tosses? What about in 9 coin tosses?

Trader Intern Interview

610

Find the lowest positive integer such that the product of its digits equals n.

Trader Intern Interview

609

There are 30 blue balls and 30 red balls, and two urns. Your opponent may arrange the balls in the two urns in any way he chooses, without telling you the arrangement. You then select one urn and draw a ball at random from it. You win $10 if you draw a blue ball, and $0 otherwise. How much should you be willing to pay to play this game?

Quantitative Analyst Interview

608

You have two decks of cards. Each deck contains both red and black cards in equal proportion. One deck has 52 cards, and the other has 104 cards, both consisting of half red and half black cards. You may choose which deck to play with. Then, you draw two cards at random from your chosen deck. If both cards are red, you win a prize. Which deck should you choose to maximize your chance of winning, and why?

Quantitative Analyst Interview

607

Design a heap data structure that supports adding elements and removing the top element, with the following constraints: (1) The topmost element must always be less than the topmost element in the left and right child subtrees; (2) The left child subtree must have as many or one more element than the right child subtree. The heap should also support a 'min' function (returns the value at the top of the tree) and an 'empty' function (returns an empty tree). Provide a description and implementation of such a data structure.

Quantitative Analyst Interview

606

Given a 4x4 chessboard, can a knight start from any square and visit every other square exactly once without revisiting any square?

Quantitative Analyst Interview

605

Suppose two players play a game where Player A and then Player B each pick an integer between 1 and 30. Then, a 30-sided die is rolled. Whoever guessed closer to the value of the roll wins an amount of money equal to the value of the roll from the other player. Given the choice, should you go first or second? What number should you choose? What is the expected value of your position?

Quantitative Analyst Interview

604

Two players each have a die. I have a 20-sided die numbered 1-20, and the other player has a 30-sided die numbered 1-30. Both players roll their respective die. If my number is greater, the other player pays me the value of my die roll in dollars. If the other player's number is greater, I pay them the value of their die roll in dollars. If we roll the same number, I pay the other player that number in dollars. What is the expected value of my winnings or losses for a single round of this game?

Quantitative Analyst Interview

603

How many heads would you expect to get if you toss 4 fair coins?

Trader Intern Interview

602

What is the probability that the sum of the numbers is even when tossing two dice?

Quantitative Analyst Interview

601

1. How many shortest paths exist from one corner of a chessboard to the opposite corner? 2. What is the smallest positive integer that has exactly 28 divisors?

Quantitative Analyst Interview

600

There are 10 castles, numbered 1 through 10, with respective values of 1 to 10 points. You have 100 soldiers to distribute among the castles in any way you choose, and your opponent does the same independently. For each castle, the player with more soldiers wins that castle's points; in the event of a tie, no one receives points for that castle. Additionally, for each castle you win, you lose 0.2 points for every soldier you have more than your opponent at that location. All 100 soldiers must be deployed. Formulate a strategy to maximize your expected score.

Quantitative Analyst Interview

599

Given that the probability it rains on Sunday is 40% and the probability it rains on the weekend (Saturday or Sunday) is 60%, what is the probability it rains on Saturday?

Quantitative Analyst Interview

597

You flip a fair coin repeatedly and stop flipping after three heads in a row occur. What is the expected number of flips required?

Quantitative Analyst Interview

598

How many handshakes occur if every person in a room shakes hands with every other person exactly once?

Quantitative Analyst Interview

596

How can you invert a pyramid of coins by moving only three coins?

Quantitative Analyst Interview

595

You are given two identical eggs and a 100-story building. Your task is to determine the highest floor from which you can drop an egg without it breaking.

Quantitative Analyst Interview

594

You are bidding on a car whose true price is uniformly distributed between 0 and 100. If your bid exceeds the actual price, you win the car and can resell it for 1.5 times its actual price. What bid maximizes your expected profit?

Internship Interview

593

What is 253 multiplied by 387? Solve this without using pen, paper, or a calculator.

Internship Interview

592

You are given a ten-sided die (values 1-10) and are allowed to roll it once or twice. After your first roll, you may choose to roll again. If you roll a second time, you add both values for your final score. If your total is 13 or less, you receive that amount in pounds as a payout. If your total exceeds 13, you receive nothing. What is the optimal strategy, and how did you arrive at your answer?

Internship Interview

590

There are 8 people in a room. Everyone shakes hands with each other exactly once. Calculate the total number of handshakes.

Internship Interview

591

There are one hundred doors, each with one dollar behind it. You roll a one-hundred-sided die one hundred times. After rolling, you may take the dollar behind the door corresponding to any number that was rolled. What is the expected amount of money you can obtain? Explain why.

Internship Interview

589

Consider a list of all the integers from 0 to 1,000,000. What is the sum of all the digits of these numbers?

Internship Interview

588

What is the expected number of heads when tossing 6 coins, given that the number of heads is greater than 2?

Internship Interview

587

You have 100 white balls and your opponent has 100 black balls. Each of you may put any number of your balls into a common pot. A third party randomly draws one ball from the pot. Whoever's ball is drawn wins an amount of money equal to the number of balls they have left over. If you know your opponent will put in 99 balls, how many balls should you put in to maximize your expected winnings?

Internship Interview

586

Player A rolls three six-sided dice (d6) and sums the values, while Player B rolls one twenty-sided die (d20). Which player has a greater probability of getting a higher number?

Internship Interview

585

What is the probability of getting an even number of heads when tossing 4 coins? How does this change with 100 coins? What if one coin is unfair? What if all coins are unfair? What if only one coin is fair? How many fair coins are needed to ensure a 50% chance of getting an even number of heads?

Internship Interview

584

What is the expected value of the result of a single fair six-sided die roll?

Internship Interview

583

What is the probability of getting an even number of heads after tossing a fair coin n times?

Internship Interview

581

What is the probability of getting exactly 2 heads in 4 coin tosses?

Internship Interview

582

One player rolls three six-sided dice (3d6), while the other player rolls one twenty-sided die (1d20). Which player has the higher probability of obtaining the greatest score?

Internship Interview

580

You have 4 fair coins. When you toss the coins, you win an amount in dollars equal to the total number of tails. Additionally, you have the option to re-toss all the coins, but this costs you 1 dollar. What is the expected value of this game?

Internship Interview

579

Six cups and saucers come in pairs: there are two red, two white, and two with stars. If the cups are placed randomly onto the saucers (one on each), what is the probability that no cup is placed on a saucer of the same pattern?

Internship Interview

578

You have a 6-sided die and a 10-sided die. You roll both dice together, guess the sum, and if you guess correctly, you win that amount in dollars. What sum should you pick to maximize your expected winnings?

Internship Interview

574

We're going to play a game. You go first. You flip a coin; if you get heads, I give you $30. If you get tails, you give me the coin and I flip. If I get heads, you give me $30; if I get tails, I give it back to you. We keep going until one of us gets heads. What is the maximum amount you would be willing to pay to go first? Give a 50% confidence range and a 90% one for your answer. Why?

Trading Intern Interview

577

What is the result of 235 minus 438?

Trading Intern Interview

576

If I roll two dice and multiply the two outcomes, what is the probability that the product is a perfect square?

Trading Intern Interview

573

You have 4 fair coins. If you flip all of them, what is the probability of getting at least 2 heads?

Trading Intern Interview

575

You play a game where you roll a 100-sided die. You can either accept the value of the roll as your payout in dollars, or pay $1 to reroll the die. What is the optimal strategy for playing this game, and what is the fair value of the game?

Trading Intern Interview

572

You have 100 blank cards and can write a single positive integer on each card. After assigning numbers, the interviewer shuffles the deck and guesses the top card. If the interviewer guesses correctly, they earn the amount written on the card. What numbers should you write on the cards to minimize the expected return of the interviewer?

Trading Intern Interview

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Jane Street

Interview Questions

Quant Interview Questions

How can you prevent gradient instability caused by the sigmoid function?

You pick n points independently and uniformly at random on the circumference of a circle. 1. What is the probability P(n) that all n points lie within some semicircle of the circle? 2. Simplify your answer to a closed form in terms of n.

How much would you pay to play a game where you roll a die and receive the number of dots rolled in dollars?

What is the probability of drawing two red cards in succession from a standard deck of 52 cards and from a double deck of 104 cards?

An unfair die with 12 faces has the number 11 with a probability of 40%, while the other faces are equally likely. You and another player are playing a game where whoever is closer to the correct answer wins. What is your optimal strategy, and should you choose to go first or second?

What is the smallest positive integer whose digits multiply to 108?

An escalator travels upwards at X steps per second. You walk up the escalator at Y steps per second, and the escalator is Z steps tall. How long will it take you to reach the top?

What is the sum of the even numbers from 1 to 60?

What is the expected number of rolls of a six-sided die to observe each face at least once? What is the expected number of rolls to observe each face at least twice?

You have 3 buckets, each able to hold one ball. Every time you throw a ball, you are guaranteed to hit a bucket; if the bucket is already filled, the ball bounces out. Each ball that goes in gives +1, each ball that bounces out gives -1. If you have 2 balls to throw, what is the expected gain?

Player 1 chooses a number, then Player 2 chooses a different number. Two dice are tossed, and whoever is closer to the outcome (the sum of the two dice) wins. Would you rather be Player 1 or Player 2? What number would you choose?

You flip a coin 4 times. What is the probability of getting exactly 2 heads?

I am thinking of a number that does not contain the digit 1. The product of its digits is 96. What is the smallest possible number? What is the largest?

It rains 30% of the time on Saturday and 40% of the time on Sunday. How often does it rain on the weekend?

If there is a 0.3 probability it will rain on Saturday and a 0.4 probability it will rain on Sunday, what is the range of possible probabilities that it will rain on at least one day?

Devise a strategy for playing a game where a fair 100-sided die is rolled, and you are paid the value shown on the die in pounds. Each roll of the die costs £1.

Find the smallest integer such that the product of its digits is 3,000,000.

What is the minimum number of people required to guarantee that there exist seven people who are born in the same month?

You roll a 6-sided die and a 10-sided die. I will pay you the sum of the numbers on the dice only if you correctly guess the sum before the roll. What number should you guess to maximize your expected payout?

You have a drawer with an infinite supply of socks in a 1:1 ratio of two colors. What is the expected number of draws needed to get a matching pair?

How many people do you need to have in a room before you can be certain that at least 5 of them were born in the same month?

How can you simulate the roll of a fair 15-sided die using a standard 6-sided die?

What is the weight of each of five melons if their pairwise weights sum to 16, 17, 18, 19, 20, 21, 22, 24, 25, and 26?

If you flip a fair coin, what is the probability of getting a head? What is the probability of getting a certain number of heads if you flip two coins? How does this generalize for n coins?

What is the probability of getting an even number of heads when tossing 4 coins?

For an odd number of fair coin flips, what is the probability of getting an even number of heads? What about for an even number of flips? How does this probability change if some of the coins are unfair?

Starting with 13 red and blue cards labeled from 1 to 13, remove 7 blue cards at random. What is the probability that a card drawn is blue, given that its number is 3?

If the chance of rain on Saturday is 70% and on Sunday is 60%, what is the probability that it will rain at least once during the weekend?

I have a number in mind such that the product of each of its digits is 96, and no digit is 1. What are the largest and smallest numbers I could think of?

What is the cube root of 169381?

Choose a subset of numbers from 1 to 30 such that no two elements in the subset share a common factor (i.e., they are pairwise coprime). What is the subset with the largest possible sum?

What is the probability of getting an even number of heads in seven coin tosses?

A train is approaching a bridge, and you are standing at the 1/4 position along the length of the bridge. If running in either direction allows you to escape just in time, what is the ratio of your speed to the speed of the train?

Two fair 6-sided dice are rolled. What is the probability that the product of the top numbers is a perfect square?

You flip a fair coin an unlimited number of times. If the sequence HHT appears before HTT, you win. What is the probability that you win?

What is the smallest number made up of only 1s and 0s that is divisible by 225?

What is the expected value of a die roll? Additionally, create a market (bid-ask spread) based on your expectation.

In Russian Roulette, there is a revolver with 4 blanks and 2 bullets loaded consecutively in adjacent chambers. If the person before you fires a blank (the gun did not discharge), should you take the next shot as-is, or spin the cylinder before shooting? Justify your answer with probabilities.

You have 4 coins. You throw them in the air. For every coin that lands heads, you get 1 dollar. How much would you pay to play this game?

What is the expected outcome when rolling two 10-sided dice? Please explain why.

What is 1,000 to the 1,000th power?

What is the minimum number of people required in a group to ensure that at least 7 people share the same birthday month?

You have two bowling balls of the same density. One has a radius of 8 and a weight of 16; the other has a radius of 12. What is the weight of the second ball?

Two players toss a fair coin repeatedly. Player A wins if the sequence HHT appears first; player B wins if HTT appears first. What is the probability that player A wins the game?

What is the expected value of an optional reroll of a fair six-sided die, given that your payout is the number of pips shown on the die? In other words, how much would you pay for the option to reroll once, given that your final payout is the number shown on the die?

Basketball players A and B each play in Game 1 and Game 2. In both games, A has a higher shooting average than B. Is it possible for B to have a higher overall shooting average than A? If so, provide an example.

A and B play a game. Each chooses a different integer between 2 and 12. Two dice are rolled, and the sum is calculated. The player whose chosen number is closer to the dice sum wins. If you can choose first or second, which position should you choose, and what is your optimal strategy?

What is the set of numbers between 2 and 30, where no two numbers share a common factor greater than 1 (i.e., the set is pairwise coprime), that gives the maximum possible sum? Using the same rules, what is the highest possible number you can have in a set of 1000?

If you randomly pick a 3-digit number, what is the probability that all three digits are even numbers?

Walk me through the solution to the birthday problem (i.e., calculate the probability that at least two people in a group share the same birthday).

How much would you pay for a game where your payoff equals the number shown on a die, with an option to reroll once? Generalize to n opportunities to reroll.

What is the probability of getting at least one head in 4 coin tosses? What about in 9 coin tosses?

Find the lowest positive integer such that the product of its digits equals n.

Given a 4x4 chessboard, can a knight start from any square and visit every other square exactly once without revisiting any square?

How many heads would you expect to get if you toss 4 fair coins?

What is the probability that the sum of the numbers is even when tossing two dice?

1. How many shortest paths exist from one corner of a chessboard to the opposite corner? 2. What is the smallest positive integer that has exactly 28 divisors?

Given that the probability it rains on Sunday is 40% and the probability it rains on the weekend (Saturday or Sunday) is 60%, what is the probability it rains on Saturday?

You flip a fair coin repeatedly and stop flipping after three heads in a row occur. What is the expected number of flips required?

How many handshakes occur if every person in a room shakes hands with every other person exactly once?

How can you invert a pyramid of coins by moving only three coins?

You are given two identical eggs and a 100-story building. Your task is to determine the highest floor from which you can drop an egg without it breaking.

You are bidding on a car whose true price is uniformly distributed between 0 and 100. If your bid exceeds the actual price, you win the car and can resell it for 1.5 times its actual price. What bid maximizes your expected profit?

What is 253 multiplied by 387? Solve this without using pen, paper, or a calculator.

There are 8 people in a room. Everyone shakes hands with each other exactly once. Calculate the total number of handshakes.

There are one hundred doors, each with one dollar behind it. You roll a one-hundred-sided die one hundred times. After rolling, you may take the dollar behind the door corresponding to any number that was rolled. What is the expected amount of money you can obtain? Explain why.

Consider a list of all the integers from 0 to 1,000,000. What is the sum of all the digits of these numbers?

What is the expected number of heads when tossing 6 coins, given that the number of heads is greater than 2?

Player A rolls three six-sided dice (d6) and sums the values, while Player B rolls one twenty-sided die (d20). Which player has a greater probability of getting a higher number?

What is the probability of getting an even number of heads when tossing 4 coins? How does this change with 100 coins? What if one coin is unfair? What if all coins are unfair? What if only one coin is fair? How many fair coins are needed to ensure a 50% chance of getting an even number of heads?

What is the expected value of the result of a single fair six-sided die roll?

What is the probability of getting an even number of heads after tossing a fair coin n times?

What is the probability of getting exactly 2 heads in 4 coin tosses?

One player rolls three six-sided dice (3d6), while the other player rolls one twenty-sided die (1d20). Which player has the higher probability of obtaining the greatest score?

You have 4 fair coins. When you toss the coins, you win an amount in dollars equal to the total number of tails. Additionally, you have the option to re-toss all the coins, but this costs you 1 dollar. What is the expected value of this game?

Six cups and saucers come in pairs: there are two red, two white, and two with stars. If the cups are placed randomly onto the saucers (one on each), what is the probability that no cup is placed on a saucer of the same pattern?

You have a 6-sided die and a 10-sided die. You roll both dice together, guess the sum, and if you guess correctly, you win that amount in dollars. What sum should you pick to maximize your expected winnings?