# 1. A hard path to enlightenment

A student at a Zen Buddhist monastery meditates for 17 hours a day. After each year of meditation, he becomes enlightened with probability 1/10, gives up and leaves the monastery with probability 2/10, and stays for another year with the remaining probability 7/10.

- What is the probability that the student eventually becomes enlightened?
- What is the expected number of years that pass before the student either becomes enlightened or leaves?
- What is the expected number of years that pass before the student becomes enlightened, conditioned on that event eventually occurring?

# 2. Two random variables

Let X and Y be independent random variables that each take on the values 1..n with probability 1/n for each value.

- Compute E[X].
Compute E[X|X>Y].

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# 3. Character generation

Thrognor the Unwashable is playing a new Massive Multiplayer On-line Roleplaying Game with a discrete mathematics theme. He rolls 3 independent six-sided dice and takes their sum to compute his *Mathematical Ability* score M.

Unhappy with the results, Thrognor selects the *Take CS202* option on the character generation screen. This deducts 197 hours from his *sleep* score, but lets him roll 4 independent six-sided dice and discard the lowest roll, giving a new *Mathematical Ability* score M'.

- What is the expected value of M?
- What is the probability that the lowest roll of four dice is equal to k, as a function of k? (Hint: First compute the probability that all four dice show at least k.)
- What is the expected value of the lowest roll of four dice?
- What is the expected value of M'?

# 4. Moving day

One day, the authorities evacuate the n occupants of a hotel with n rooms. When the occupants return, they are each assigned a new room according to a uniform random permutation. Use Chebyshev's inequality to show that the probability that more than k occupants end up in their old rooms is at most 1/k².