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I heard / read / acquired this puzzle from somewhere:

Roll a die, win $2\times$ the value of the face if it is an even number, or win $1\times$ the value of the face if it is an odd number.

How much do you need to charge to play the game?

Then if you are given the opportunity to make a decision and play again, you will only be paid by the result of the last rolling.

Again, how much do you need to charge to play the game?

I understand the first part (pretty easy) and I got 5.5.

I don't really get the second part.  According to the puzzle’s source, the correct answer is 7.  Can anyone please go through the steps to getting 7 in detail please?

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    $\begingroup$ Dice is plural -- one die, two dice. $\endgroup$
    – Glen_b
    Sep 3, 2017 at 2:12
  • $\begingroup$ I interpreted "play again" not as "re-roll", but as "pay again to re-roll". The question could be a bit more precise...unless the task is to figure out the meaning of the question (which is, in that case, possible since the correct answer to part 2 is given). $\endgroup$
    – Klaws
    Sep 3, 2017 at 9:23
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    $\begingroup$ You should document the source of this puzzle.   Don’t do it in a comment; edit your question. $\endgroup$
    – Peregrine Rook
    Sep 5, 2017 at 4:23

1 Answer 1

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How I interpret that condition is that the player has an opportunity to ignore the first result and take a second chance, but they must use the second chance if they do reroll. Possible prize values are 1,4,3,8,5,12.

Anyone playing logically would take the reroll option if they got something lower than the expected value of the reroll (5.5). Therefore, they would reroll when rolling 1, 2, 3, or 5 on the first roll, which gives 1, 4, 3, and 5 prizes. On the initial roll, there is a 1 in 3 chance of the player getting a 8 or 12 prize, expected value of 10. On the other 2 in 3, they will take the reroll option and on average get 5.5. Adding the weighted expected values of these results will give the expected value of a single game.

$(10*1/3) + (5.5*2/3)=7$

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  • $\begingroup$ Thank you so much. Now I understand. At first, I didn't understand how you get 5.5 in the sentence 'On the other 2 in 3, they will take the reroll option and on average get 5.5'. I completely ignored the original expected value 5.5.. which will stay the same in the 2nd reroll. $\endgroup$
    – JungleDiff
    Sep 3, 2017 at 1:15

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