You are presented with two empty jars and 100 marbles
on a table. There are 50 white marbles and 50 black marbles. You are
to put all 100 of the marbles into the two jars in any way you choose. I
will then blindfold you. I will shake the jars up to ensure good mixing,
and I will rearrange the placing of the jars on the table so that you do
not know which one is which. You may then request either the “lefthand”
or the “right-hand” jar. You get to choose exactly one jar, you
are allowed to withdraw at most one marble from the jar, and you do
not get a second chance if you are unhappy with your choice.
How many of each colour marble should you place in each jar to maximize
the probability that your blindfolded random draw obtains a white
marble?

How many 0s at end of 100!

3m + 7n = 10000, find all such pairs of positive integers

Calculate value of barrier option for 3-period binomial model such that $u = 0.1, d = -0.1, r = 0.05, K = 90, B = 100, S_0 = 100$ with payoff: $(S_T - K) * I(S_T < B)$.

Here is a simple game. You get to toss a fair coin now. If it is heads, you get seven dollars 18 months from now. If it is tails, you lose two dollars immediately. The one-year interest rate is 12% per annum. The two-year interest rate is 18% per annum. How much are you prepared to pay to play this game?

Player M has 1 dollar and player N has 2 dollars. Each game gives the winner 1 dollar from the other. As a better player, M wins $2/3$ of the games. They play until one of them is bankrupt. What is the probability that M wins?

Two players bet on roll(s) of the total of two standard six-face dice. Player A bets that a sum of 12 will occur first. Player B bets that two consecutive $7$s will occur first. The players keep rolling the dice and record the sums until one player wins. What is the probability that A will win?