1.1

What is the value of $\sqrt{2}^{\sqrt{2}^{{\sqrt{2}^{...}}}}$?

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3.1

You have a bag with two coins. One will come up heads 40% of the time, and the other will come up heads 60%. You pick a coin randomly, flip it and get a head. What is the probability it will be heads on the next flip?

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3.3

In front of you is a jar of 1000 coins. One of the coins has two heads, and the rest are fair coins. You choose a coin at random, and flip it ten times, getting all heads. What is the probability it is one of the fair coins?

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3.5

Starting at one vertex of a cube, and moving randomly from vertex to adjacent vertices, what is the expected number of moves until you reach the vertex opposite from your starting point?

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3.8

You have a spinner that generates random numbers that are uniform between 0 and 1. You sum the spins until the sum is greater than one. What is the expected number of spins?

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3.10

A stick is broken randomly into 3 pieces. What is the probability of the pieces being able to form a triangle?

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3.11

A stick is broken randomly into two pieces. The larger piece is then broken randomly into two pieces. What is the probability of the pieces being able to form a triangle?

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3.12

This is based on a Goldman Sachs interview question. You play a game where you toss two fair coins in the air. You always win $1. However, if you have tossed 2 heads at least once, and 2 tails at least once, you surrender all winnings, and cannot play again. You may stop playing at anytime. What’s your strategy?

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5.2

You have a basket of $n$ assets. The asset returns are multivariate normal with zero mean. Correlation between any pair of assets is 1/2. What is the probability that $k$ of the assets will have positive return?

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5.10

Suppose there are 36 students in the QFRM program, each assigned a unique integer from 1 to 36. Thirty-six quarters are laid out on a table in a row, heads up. Each student goes to the table, and if they are assigned the number $n$, they turn over the $n$th coin, the $2n$th coin, and so on. So, for example, the student who is assigned 15 will turn over the 15th and 30th coins. When everyone is done, how many tails are showing?

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