1) $1

2) infinity (well, expectation is infinity)

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Quantitative Analyst Interview New York, NY

Knight CapitalAnswer## How much would you pay to play this game? You have a fair

coin. You get heads you win $1 and can continue to play. You get tails, you collect your winnings ans stop the game. Second question. Now what if the winnings double each time you get heads.

4 Answers

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1) $1

2) infinity (well, expectation is infinity)

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The fair amount to be paid for this game is the expected winnings from it. Let x be the expected winnings. Now, to calculate x, consider the first toss - With probability 1/2, we get a tail thus earning $0. With probability 1/2, we get a head to get $1 and make another $x in expectation from further tosses.

x = 1/2 * 0 + 1/2 * (x + 1) => x = 1

In the second scenario, if winnings double with every toss:

y = 1/2 * 0 + 1/2 *( 2 * y + 1) => y = infinity (ie expectation diverges)

where y is the expected winning.

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After simulating the second scenario a bunch of times on Python (each averaging 1,000,000 iterations), there was an average payoff roughly in the range of 4-12. Obviously, the expectation is infinite, but in practice it is reasonable to use monte carlo estimations for a more realistic answer. Indeed sometimes you can win massively, sometimes you can win nothing.

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simple expectation formula. For example, for 4 trows . E(Reward) = 0.5*0 + (0.5*1 + 0.5* ( 0.5 *2 + 0.5*(0.5*3 + 0.5 * 4)))) = ... The sum converges if you continue the formula, for example to 10 trows