Problem 1.

What is the next date whose digits are all unique?

Problem 2.

If there are $n$ people in the party, is it true that there will be two with equal number of friends in this party?

Problem 3.

How many 3-digit numbers of the form $\overline{abc}$ are there such that $a > b > c$?

Problem 4.

Eight quants from different banks are getting together for drinks. They are all interested in knowing the average salary of the group. Nevertheless, being cautious and humble individuals, everyone prefers not to disclose his or her own salary to the group. Can you come up with a strategy for the quants to calculate the average salary without knowing other people’s salaries?

Problem 5.

Eleven pirates looted a chest full of 100 gold coins. Being a bunch of democratic pirates, they agree on the following method to divide the loot:
The most senior pirate will propose a distribution of the coins. All pirates, including the most senior pirate, will then vote. If at least 50% of the pirates (6 pirates in this case) accept the proposal, the gold is divided as proposed. If not, the most senior pirate will be fed to shark and the process starts over with the next most senior pirate... The process is repeated until a plan is approved. You can assume that all pirates are perfectly rational: they want to stay alive first and to get as much gold as possible second. Finally, being blood-thirsty pirates, they want to have fewer pirates on the boat if given a choice between otherwise equal outcomes. How will the gold coins be divided in the end?

Problem 6.

Calculate the probability of getting at least 2 heads after 4 coin flips. What about 100 coins and at least 50 heads?

Problem 7.

There are 50 white and there are 50 black balls, which you distribute into two sacks in any way you want (but none of the sacks can be empty). Then I blindly pick a sack and choose a ball from it (also blindly). If it's a black ball, then I win. Otherwise you win. How many white and how many black balls will you put in the first sack (the rest will go to the second one)?

Problem 8.

You have $2n$ vertical ropes in a row of the same length, but you can see only ends of each rope (so you don't know which ends on the top correspond to which ends on the bottom). You split the top ends of ropes into pairs, and connect them (those top ends in each pair). Then you do the same for the bottom ends of the ropes. Find the probability to get one single loop.

Problem 9.

What is the probability that the product of two randomly chosen random numbers from the interval $[0,1]$ is gonna be less than a half?

Problem 10.

Two points are uniformly chosen on a stick (independent of each other). These two points are the points where the stick is gonna be cut. What is the probability that the 3 resulting segments can form a triangle?

Problem 11.

A company is holding a dinner for working mothers with at least one son. If you meet Ms. Jackson, a mother with two children, at that party, what is the probability that both of her children are boys (you know she has two children exactly)?

Problem 12.

Jason throws two darts at a dartboard, aiming for the center. The second dart lands farther from the center than the first. If Jason throws a third dart aiming for the center, what is the probability that the third throw is farther from the center than the first? Assume Jason's skillfulness is constant.

Problem 13.

There's a treasure box with 100. There's a 50% probability that the box contains a bomb. The bomb has a probability of exploding on the $i$-th (for $i \leq 100$) day given by a uniform distribution. The bomb will eventually explode if there is a bomb. If it does not explode in the end, you can take all the money inside the treasure box. On the $i$-th day, how much are you willing to pay to play the game? (you do not die if the bomb explodes, but the game stops)

Problem 14.

Two people flip coins independently. The probability of getting a head is $p$. A flips first. What is the probability that A gets two consecutive heads earlier than B?

Problem 15.

You are taking out candies one by one from a jar that has 10 red candies, 20 blue candies, and 30 green candies in it. What is the probability that there are at least 1 blue candy and 1 green candy left in the jar when you have taken out all the red candies?

Problem 16.

What is the expected number of rolls of a fair die until the first time you see three consecutive ones?

Problem 17.

Getting heads-tails-heads and heads-heads-tails are equiprobable after 3 coin flips. But if I keep flipping a coin, I'm more likely to get one of these combinations than the other. Which one? Why?

Problem 18.

You and Bob are playing a game. It all starts with each of you making a bid of some integer number of coins (you don't tell each other your bid). In case if your bid is larger or equal to Bob's, you win this "auction", you pay whatever you bid and play the following game: a fair coin is filled 200 times, every coin that is tossed Heads is the coin you get (you do not get any other coins). E.g if you won the auction with a bid of 67, and if in the tossing-a-coin game there were 93/200 "Heads", you will win $93-67=26$ coins in total. What would be your strategy?

Problem 19.

There are $n$ cars on a one-lane road all going in the same direction. Initially they are all at a large distance from each other. Suppose their velocities are distributed uniformly on [0, 1]. When a faster car reaches a slower one from behind, the slower car feels really bad and pulls out from the road. Given that a long enough time passes, find the expected number of cars that will remain. E.g, if there are only two cars on the road, the expectation is $0.5 \cdot 2 + 0.5 \cdot 1 =1.5$.

Problem 20.

There are 10 distinct types of coupons in cereal boxes and each type, independent of prior selections, is equally likely to be in a box.
a) If a boy called Darsh wants to collect a complete set of coupons with at least one of each type, how many boxes on average are needed to be opened by him?
b) If Darsh has collected $n$ coupons, what is the expected number of distinct coupon types he has?

Problem 21.

What is the expected number of cards that need to be turned over in a regular 52-card deck in order to see the first ace?

Problem 22.

You have a fair 6-sided die with numbers from 1 to 6. You start with a score of 0, and every time you roll the dice you add the resulting number to your current score. Once you get at least a 100 as your score, you stop. Estimate the expected value of your score.

Problem 23.

A bowl contains $N$ spaghetti noodles. You reach into the bowl and grab two free ends at random and attach them. You do this $N$ times until there are no free ends left. On average, how many loops are formed by this process?

Problem 24.

There are $n$ couples who went to a dancing session. In each of them the husband is of exactly the same height as the wife. The moment music starts playing, people will split randomly into pairs "some husband and some wife". Find the mathematical expectation of the number of pairs in which the husband is taller than the wife he is dancing with.

Problem 25.

A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they cannot have twins, and that each child is a boy-girl with 50-50 chance. What is the expected number of children they will have?