If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

Suppose that we have a row of \(5\) taps, all of which run into the same tank. Each tap fills the tank at a different rate, such that the \(n\)th tap can fill the tank in \(n\) hours.

Two taps, chosen at random, are turned on and allowed to fill the tank. The expected time, in hours, that it will take to fill the tank is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a - b\).

Note: You may need a calculator for the last step of the problem.

Jenny places \( 100 \) pennies on a table, with \( 30 \) showing heads and \( 70 \) showing tails. She chooses \( 40 \) distinct pennies uniformly at random and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. After this process, what is the expected number of pennies showing heads?

This problem is shared by Muhammad A.

You are on an infinite triangular lattice with black circles repeated as shown below, and you start at the vertex circled in red:

Every move you randomly walk along a black line segment to a neighboring vertex.

What is the expected value for the number of moves before you hit one of the lattice points with a black circle on it?

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