Killer Interview Question: How Much Money Would You Want To Run This Betting Game?

Another addition to the killer interview questions collection: how much would we have to pay you to run this betting game?

Betting picture from Shutterstock

More specifically, the question (used by Goldman Sachs) involves the following game:

The rules are that I flip a coin, and if it comes up heads, you pay me a dollar and the game is over. If it comes up tails, you flip again. If it comes up heads the second time, you pay me two dollars, and the game is over. If it comes up tails again, you flip again. Third time, you pay me four dollars for heads and the game is over, and you flip again for tails. And so on and so on, each time doubling the payout for heads, and flipping again on tails. How much would I have to pay you up front to play this game?

The question is designed to both tests your mathematics knowledge and assess your appetite for risk -- a relevant consideration for security professionals as well as investment bankers. As this analysis explains, the odds of your paying out more than $8 are very low, so that might be a safe number. But I'm inclined to think the better answer is "I wouldn't take part in something where there was a potential for a really massive loss".

How would you answer the question?

Goldman Sachs Takes One Of Its Interview Questions From An 18th Century French Betting Game [Business Insider]


    Felix salmon has it right - Before you start playing, at each coin toss your expected payout is 2^(n-1)*0.5^n - which simplifies to 50c. So the value of the game as a whole is infinite! If you were to play this game often enough, you would be guaranteed to lose money no matter how much you were paid up front.

    There's a more interesting answer to the question if you bring in the idea of bankruptcy and/or bailouts.

    In that case, if you have assets of (say) $100k, beyond which you go bankrupt, then it makes sense to play - you can't lose more than that $100k (because your debts just get wiped out in bankruptcy) - or if you are in Goldman's position, it's even better because the government will cover the tab!

    This is the source of the IBGYBG (I'll Be Gone, You'll Be Gone) mentality was pervasive before the GFC - keep playing the martingale, because if heads comes up 40 times in a row, I'll be gone and you'll be gone, so keep playing!

    "How much would I have to pay you up front to play this game?"

    Nothing is paid up front to play the game.

    "The rules are that I flip a coin, and if it comes up heads, you pay me a dollar and the game is over."

      Fail. You missed the rest of the rules. You can't just stop reading at the first one.

    The odds of winning Powerball are around 76 Million to 1 (for a single game). Let's go with 76 Million as the number of coin toss games played. The expected payout (rounding down) is around $973M. This requires a $12.80 play fee to break-even. If someone got lucky and managed a 27 toss game though you'd be down $67 Million (less $12.80).

    Doing a quick simulation with just 523,000 games gave me a result of $9.50 for break-even. There were 3 games with 18 tosses paying out $131k each. Given the nature of the problem the more games played the larger you're buy-in needs to be.

    If there's only 1 person playing then it isn't worth it to play. You're risking a lot for a small potential gain. As a business though it would be a different story. Incidentally, the Martingale System is part of the reason why casinos have a maximum bet on roulette tables: otherwise a deep-pocketed gambler could keep doubling down on red/black (even though the odds aren't quite 50:50, this would word with infinite bank).

    The question isn't how much you would have to pay me to play this game, the question is, after you've run the game, do I get to run it in reverse with each member of the interview panel?

    Pay me $32. I show a loss if there more than 5 tails in a row.

    If we play long enough, such a run will eventually happen. The things is, paying $32 gives you ONE run at my money, and the odds of getting 5+ tails in a row, straight off the bat, are quite low.

    The kicker is: we only play ONCE. If you want me to play a second time, you need to give me another $32. Going by that rule, I should be well in front by the time that 5+ streak comes around.

      There is a chance, however small, that you will pay out more than $32 in the first game.

      The odds of getting 5+ tails in a row straight off the bat are exactly the same as getting 5+ tails in a row in any future game.

      Put it this way - at Jupiter's casino, one croupier spun red on the roulette table for their entire shift. Are you prepared for that kind of loss?

        While it is theoretically possible, I'm gonna have to call bullshit on that. An entire shift with, lets say a minimum of 6 1/2 hrs dealing (and assuming they are only dealing roulette), with even a low-moderate spin rate would be in the range of 130-180 spins. Even if only half their swing was roulette you are still looking at 60-90 spins for a shift. A single hour on a table spinning 20-25 in a row would be much more believable. My personal record is 18.

          It happened - my uncle was the pit boss at the time. And a shift lasts for much less than 6.5 hours. Still, the croupier in question was getting worried about keeping her job.

            6.5hrs of dealing over an 8hr shift is pretty normal assuming tables aren't closing, unless you somehow don't mean their entire shift of dealing but simply one table of their swing which is usually 1-1.25hrs (sometimes as short as 40min at times) depending on casino. If it is the case of a single table then yes, that is within reason as I already pointed out above...

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