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  1. 053You roll a fair die and are paid its face value in lakh rupees. After seeing the first roll you may roll once more and take the second result instead, whatever it turns out to be. What is the game worth, and what is your stopping rule?Expected value and gamesCoreBulge bracket IBConsulting style brainteasers

    Try it first

    On which first rolls do you roll again?

    Show the worked solution

    The game is worth Rs 4.25 lakh, and you roll again on a 1, 2 or 3. A single roll is worth 3.5 lakh on average, so the chance to roll again is worth exactly that. Keep any first roll above 3.5, which means 4, 5 or 6. Half the time you keep, averaging 5 lakh; half the time you roll again for 3.5. Half of 5 plus half of 3.5 is 4.25 lakh. The option to re-roll adds 0.75 lakh over a single roll.

    Why is the rule to compare against 3.5?

    Think of boarding a train with one visible seat by the door and the rest of the carriage out of sight. You walk on only if the seat you can see is worse than the average seat down the carriage, because once you walk you take whatever is there. Work backwards: the last decision has no choices left, so its value is the plain average, 3.5 lakh, and any earlier roll is kept only if it beats that. A 4, 5 or 6 beats 3.5; a 1, 2 or 3 does not; no face equals it, so the rule has no ties. The number you are comparing with is the expected valueThe average outcome if the same gamble were repeated many times: each result weighted by its probability. of rolling again.

    Keep a 4, 5 or 6; re-roll a 1, 2 or 3: the bar is what rolling again is worthFirst rolleach face one in six1re-rollworth 3.52re-rollworth 3.53re-rollworth 3.54keepworth 45keepworth 56keepworth 63.5: the value of rolling againSecond roll: no choice left,take whatever shows, average 3.5Kept rolls contribute(4 + 5 + 6) / 6 = 2.5Value = 2.5 from the kept rolls + 3/6 x 3.5 = 1.75 from the re-rolls= 4.25 lakhKeeping a 3 as well, or re-rolling a 4, both give 4.17: the 3.5 bar is the only one that reaches 4.25.
    A first roll of 1, 2 or 3 falls below the 3.5 that a second roll is worth, so you roll again, while a 4, 5 or 6 is kept, and the kept faces contribute 2.5 and the re-rolls 1.75 for a game worth 4.25 lakh.

    How does the 4.25 come together?

    Three faces out of six are kept and contribute (4 + 5 + 6) over 6, which is 2.5 lakh; three are re-rolled and contribute three sixths of 3.5, which is 1.75; the total is 4.25 lakh. Check the rule by moving it. Keep the 3 as well: (3 + 4 + 5 + 6) over 6 plus two sixths of 3.5 is 4.17. Re-roll the 4 too: (5 + 6) over 6 plus four sixths of 3.5 is 4.17. Both sit below 4.25, about Rs 8,333 a game, so 3.5 is the only bar that gives the full value.

    The relationship
    V=4+5+66+36×3.5=2.5+1.75=4.25V = \frac{4 + 5 + 6}{6} + \frac{3}{6} \times 3.5 = 2.5 + 1.75 = 4.25
    (4 + 5 + 6)/6the kept faces, each one in six
    3/6the chance the first roll is a 1, 2 or 3 and you roll again
    3.5what the second roll is worth on average
    What it says in wordsAdd what you keep, weighted by its chance, to what you get from rolling again, weighted by its chance.

    What happens with a third roll, and why do interviewers ask?

    Allow two re-rolls. The game from the second roll onward is now worth 4.25, so on the first roll you keep only a 5 or 6: (5 + 6) over 6 plus four sixths of 4.25 is 4.67 lakh. Every extra chance to try again raises both the value of the game and the bar for stopping, and that is the whole logic of an option: the right to swap a known result for an uncertain one is worth paying for only when the known result is poor. The same reasoning prices walking away from a signed deal or waiting for a better bid. The limit: this assumes you care only about the average. Someone who badly needs 3 lakh might keep a 3, and that is a preference, not an error.

    Where candidates lose it

    The common miss is a rule based on feel, such as re-rolling anything below 5 because 5 and 6 feel like wins. That throws away a 4 that beats the 3.5 you get by rolling again, and costs about Rs 8,333 of value per game.

    The second loss is valuing the game at 3.5 lakh because it is still just a die. The option to re-roll is worth 0.75 lakh here, and pricing an option at zero is the error the interviewer is listening for.

    What the interviewer asks next

    • You may re-roll twice. What is the game worth and what is the rule on the first roll?
    • The re-roll costs Rs 50,000. Do you still take it, and when?
    • How does this connect to the decision to walk away from a signed deal?