Probability puzzles




Medium Probability

The stick drops and breaks at a random point distributed uniformly across the length. What is the expected length of the smaller part?

Solution
Source: Quant Interview
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Hard Probability

There are 26 black(B) and 26 red(R) cards in a standard deck. A run is number of blocks of consecutive cards of the same color. For example, a sequence RRRRBBBRBRB of only 11 cards has 6 runs; namely, RRRR, BBB, R, B, R, B. Find the expected number of runs in a shuffled deck of cards.

Hint
Answer
Solution
Source: Quantnet
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Hard Probability

An ant is standing on one corner of a cube & can only walk on the edges. The ant is drunk and from any corner, it moves randomly by choosing any edge! What is the expected number of edges the ant travels, to reach the opposite corner?

Hint
Solution
Source: Quant Interview
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Hard Probability

What is the expected distance of any point on Earth and the north pole? Take Earth radius 1.

Clarification: Shortest distance cuts through the sphere, instead of lying on surface.

Further thinking: Is this question same as choosing two random points on unit sphere and asking their expected distance?

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Answer
Solution
Source: Quant Interview
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Hard Probability

A stick is broken into 3 pieces, by randomly choosing two points along its unit length, and cutting it. What is the expected length of the middle part?

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Solution
Source: Quant Interview
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