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Discrete Probability

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.

Level 2

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?

Alex tells the truth 70% of the time while Blake tells the truth 80% of the time, independently of each other. They are shown an apple and asked "Is it an apple or a banana?". What is the probability that they will give different answers?

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

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 \(\alpha + \beta\).

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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