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How often will a die come up "4"? How likely is it to rain tomorrow? Probability is one of the most powerful frameworks for modeling the world around us.

There are 100 people in line to board a plane with 100 seats. The first person has lost his boarding pass, so he takes a random seat.

Everyone that follows takes their assigned seat if it's available, but otherwise takes a random unoccupied seat. What is the probability the last passenger ends up in his/her assigned seat, as a decimal?

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In a convex hexagon, two diagonals are drawn at random. What is the probability that the diagonals intersect **inside** the hexagon?

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Two players, Nihar and I, are playing a game in which alternate tossing a fair coin and the first player to get a head wins. Given that I toss first, the probability that Nihar wins the game is \(\frac{\alpha}{\beta}.\)

If \(\alpha\) and \(\beta\) are coprime positive integers \(\alpha\) and \(\beta\), find \(\alpha + \beta\).

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Out of four machines, Titus knows that exactly two are faulty. He wants to test them one by one in a random order (without replacement) until he can identify the 2 faulty machines. Find the probability that exactly two tests are needed?

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