Determining Optimal Strategy for Rolling Dice

[bob j]

Hi All,
I was wondering if anyone could give me a suggestion to solve this

"With one die, suppose in a round, you earn the amount of dollars equal to the value of the upwards face of the die. (eg. you earn $6 if you roll a six.) Now also suppose after your first roll, you are given the opportunity to cancel your first and roll again, taking that value as the final value. What should your strategy be? What if you are allowed to roll the die three times?"

thank you,

 

[CRGreathouse]

What are you expected winnings on a single roll of the die? Clearly this is $3.50, since all results 1-6 are equally probable.

If you get to roll a second time, you should take it if and only if your expected winnings are higher by rolling again -- that is, if you roll less than 3.5. This happens precisely when you roll a 1, 2, or 3, so your strategy is to roll again in precisely these cases. This makes your average winnings $(3.5 + 3.5 + 3.5 + 4 + 5 + 6) / 6 = $4.25.

If you get to roll three times, you should only take the first reroll if you could get better than 4.25.

 

[ych22]

Hi All,
I was wondering if anyone could give me a suggestion to solve this

"With one die, suppose in a round, you earn the amount of dollars equal to the value of the upwards face of the die. (eg. you earn $6 if you roll a six.) Now also suppose after your first roll, you are given the opportunity to cancel your first and roll again, taking that value as the final value. What should your strategy be? What if you are allowed to roll the die three times?"

thank you,

If the first throw is a number from {1,2,3}, then your expected winnings if you roll again is $3.5 so you should cancel and roll again.

If you are allowed to throw 3 times and take the maximum from those 3 (let it be Y), you should observe that Y follows the following discrete distribution:

P(Y=1)=1/216
P(Y=2)=7/216
P(Y=3)=19/216
P(Y=4)=37/216
P(Y=5)=61/216
P(Y=6)=91/216

E(Y)= 4.96

Your strategy should be to re-roll if your first roll was from {1,2,3,4}.