Discrete Mathematics
# Expected Value

A game costs $150 to play. In this game, you roll a fair six-sided die until all six numbers have been rolled at least once. You are then paid 10 times the number of rolls you made.

For example, if the rolls were 3, 5, 4, 3, 2, 5, 1, 4, 1, 3, 6, then you would get \((10)(11) = 110\) dollars.

Including the price to play, what is your expected value in this game?

For fixed integers \(n\) and \(k\), find all \(k\)-tuples of non-negative integers \( (a_1, a_2, \ldots a_k) \) such that

\[ a_1 + a_2 + \cdots + a_k = n \]

For each integer \( i \) from 0 to \(n\), let \( p_i \) be the probability that one of the \( a_j \) is equal to \(i \). What is the value of

\[ 1\times p_1 + 2 \times p_2 + \cdots n \times p_n ? \]

Also try Daniel Liu's Expected Distance on a Circle.

A square \(PQRS\) is inscribed inside a square \(ABCD\), with \(P \) on \(AB\). If \(P \) is uniformly distributed along side \(AB \), the expected area of \(PQRS\) is \(24\text{ unit}^2\). What is the side length of square \(ABCD\) (in \(\text{unit}\))?

Give your answer to 1 decimal place.

An ant starts on one vertex of a dodecahedron. Every second he randomly walks along one edge to another vertex. What is the expected value of the number of seconds it will take for him to reach the vertex opposite to the original vertex he was on?

**Clarification:** Every second he chooses randomly between the three edges available to him, including the one he might have just walked along.

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