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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.

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.

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A bag initially contains one red ball and one blue ball. You select one of these balls at random and then place two balls of that same color back in the bag. For instance, if you drew a red ball, then you would put two red balls back into the bag so that the bag would contain two red balls and one blue bag. You repeat this until the bag contains 2016 balls. For the color that has the most balls present in the bag, what is the expected number of balls of this color? That is, after the bag contains 2016 balls, you open it and choose the color that has more balls (or choose either color in case of a tie); what is the expected number of balls of this color?

If the answer can be expressed as \(\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, enter \(p+q\) as your answer.

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In India, there is an exam **JEE-Mains** which is taken by more than a million students. Here, there are 90 objective problems, each having 4 choices, out of which only 1 is correct. Each correct answer carries 4 marks, and each incorrect answer costs -1 marks. You get 0 marks for an unattempted problem.
###### Image credit: The Hindu

Consider a very special student sitting in the exam. This student knows at least 1 of the problems correctly. The probability that he knows the correct answers to exactly \(k\) problems is directly proportional to \(\frac{1}{k}\). In each of the problems he doesn't know the answer to, there is an 80% chance that he marks it randomly and a 20% chance that he leaves it unanswered.

Find the greatest integer less than or equal to the expected marks of this student.

**Details and assumptions**

\(\displaystyle \sum_{n=1}^{90} \frac{1}{n} = \frac{1}{0.196751}.\)

If he answers a problem randomly, the probability that he gets it correct is exactly \(\frac{1}{4}.\)

As any other student, if he knows the correct answer to a problem, he will mark it, i.e. he won't leave it unanswered.

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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?

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I toss a fair coin 100 times in a row. If the expected number of times that two Heads are tossed consecutively is equal to \(N\), what is the value of \(100N\)?

**Clarification**: A run of three or more Heads in a row will contribute more than one instance of consecutive Heads. If three Heads are tossed in a row, then two Heads have been tossed consecutively twice. If six Heads are tossed in a row, then two Heads have been tossed consecutively five times, and so on. For example, two Heads have been tossed consecutively 11 times in the following sequence of 30 tosses:

\[ THHTHHHHTHHHTHTHHHHHTTTTHHTHTT \]

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