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# Expected Value

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.

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.

A point is chosen randomly (by distribution of area) on the inside of a unit circle. Find the expected value of its distance from the center of the circle.

If you get your answer as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, submit your answer as \(a+b\).

One day, you decide to play a game with your friend out of sheer boredom. Your friend picks out an integer randomly from the range \([1,1023]\) and you try to guess it.

You can guess an infinite number of times. Let \(n\) denote the integer your friend chose and let \(k_i\) denote the integer you chose on the \(i^\text{th}\) try. If \(n > k_i\) your friend will tell you "higher", if \(n < k_i\) your friend will tell you "lower" and if \(n = k_i\) then you win the game.

But then, your friend decides this is too boring. He decides to give you 1023 dollars. Every time you make a guess, right or wrong you have to give him \(x\) dollars, even if you already gave him back the original 1023 dollars or even if you already have negative profits. Since your friend knows that you will play optimally, he sets \(x\) such that your expected profit after playing the game is 0. Find \(\lfloor x \rfloor\).

**Notation**: \( \lfloor \cdot \rfloor \) denotes the floor function.

You have an **unfair** coin. The probability it will land on heads is \( \dfrac23\).

The expected value for the number of coin flips it will take to get 3 heads in a row can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

What is \(a+b\)?

Other Expected Value Quizzes

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