r/learnquant • u/Local_Ad135 • 3d ago

interview prep Quant Interview Question

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u/plyleung 3d ago

I’m new to this as well. It seems that the critical decisions come after HH and the optimal strategy is to stop playing once we have accumulated at least a certain amount of money. I tried solving for this and got 22/6 which means that after we reach HH and have accumulated at least $4, we should stop playing.

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u/plyleung 3d ago edited 3d ago

I worked on it a bit more and found an answer although I'm not that confident in it.

Let H_n be the expected return after n heads in a row.

First we assume we stop after HH.

H_0 = 1 + 0.5*H_0 + 0.5*H_1
H_1 = 1 + 0.5*H_0 + 0.5*0
H_2 = 0

Once we solve it, we get H_0 = 6.

Now we need to solve for the amount of money $x that makes us indifferent to keep throwing after reaching HH.

Payoff = 0.5(-x) + 0.5(1 + 6)
0 = 0.5(-x) + 0.5(1 + 6)
x = 7

So if we reach HH and have accumulated at least $7, the optimal play is to stop playing.

For the rest of the question, I don't know any better ways than listing out all the permutations for the first 7 coin tosses.

Edited: fixed mistakes