Probability puzzles

What is the expected number of coin tosses required to get n consecutive heads?

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Solution
Source: Top Quant Interview
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A chess tournament has K levels and 2^K players with skills 1 > 2 > ... >2^K. At each level, random pairs are formed and one person from each pair proceeds to next level. When two opponents play, the one with better skills always wins. What is the probability that players 1 and 2 will meet in the final level?

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A coin is tossed 10 times and the output written as a string. What is the expected number of HH? Note that in HHH, number of HH = 2. (eg: expected number of HH in 2 tosses is 0.25, 3 tosses is 0.5)

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

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Source: Quant Interview
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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.

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Source: Quantnet
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