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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.

Level 5

         

Assuming that the earth is a perfect sphere with radius 6378 kilometers, what is the expected straight line distance through the earth (in km) between 2 points that are chosen uniformly on the surface of the earth?

Suppose the Cookie Monster has two (distinguishable) jars of cookies, each of which initially contains \(20\) cookies. The Monster then starts to eat the cookies one at a time such that each time he takes a cookie from a jar chosen uniformly at random.

Now the Cookie Monster never looks into the jars, so he never knows how many cookies are left in either jar until he reaches in to one and finds that that jar is empty. When he first discovers that one of the jars is in fact empty, the probability that there are (strictly) fewer than \(3\) cookies in the other jar is \(S.\)

Find \(\lfloor 1000S \rfloor\).

A square with a unit length has two arbitrary points chosen inside it. What is the average distance between the two points?

If this can be represented in the form \(\dfrac {a + \sqrt{b} + c \ln (d+\sqrt{e})}{f}\), where \(a, b, c, d, e, f\) are positive integers, and the fraction is simplified as far as possible, find \(a+b+c+d+e+f\).

A row of street parking has spaces numbered 1 through 100, in that order. Each person trying to park drives by the spaces one-by-one starting at number 1.

At each open space, they are told how many open spaces, \(S,\) they haven't passed (including the current space). They park in that space with probability \(\frac{1}{S};\) otherwise, they move on to the next open space.

What is the expected value for the parking spot of the \(100^\text{th}\) person to arrive (assuming no one has left and only one car goes through the parking process at a time)?

In my favorite game, I start with 3 lives. In each round, I have a 10% chance of dying, losing a life. If I don't die, I am given an extra life, but I cannot ever have more than 3 lives.

What is the expected value for the number of rounds I will play before losing the game (when I lose my final life)?

Note: Each round is counted, even if you lose a life in the round.


Inspiration.

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