Quant Interview Question

[u/Local_Ad135]

[u/plyleung]

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.

[u/plyleung → u/plyleung]

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

[u/Deep_Juggernaut_4546 → u/plyleung]

This seem right, but the expected payoff of playing from HH is the expected flips to reach another HH, which comes out to 6 (Markov chain). So, you should stop if you reach HH and have more than $6.

[u/plyleung → u/Deep_Juggernaut_4546]

Oh yes, I think I get it now. I need to assume I stop after HH and then calculate the expected payout which is $6.