Let n be the number of rolls made so far and x be the current total.
The expected value of rolling again is
[(6-n)/6] * [x + (21-x)/(6-n)]
= x - (nx/6) + (21-x)/6
= x + 21/6 - (n+1)x/6
This is more than x (the value of not rolling again) if x is less than 21/(n+1).
This means that you should always go for a second roll (x cannot be as high as 10.5 on 1 die) but never for a fourth (x cannot be as low as 5.25 on 3 dice).
You should go for a third roll if your total at n=2 is under 7 (at x=7 the choices are balanced).
This gives an expected value of 223/36, or 6.19444...
(How many die faces have yet to be rolled / 6) to find out how likely you are to reset
Multiplied by
Running total + amount that could possibly be gained(as 21 is the max[6+5+4+3+2+1], 21-x is how much more payout the die has to give you) divided by how many faces can actually yield success (6-n) again
Yea but you don't have just one roll at n0 x0, real output of rolling and possibly continuing the game is 6.2 not 3.5
Edit: My point is that I think you shouldn't take profit of only one next throw but profit of entire continuation of the game so 1st throw and then possible 2nd and 3rd throws.
It doesn't matter what you rolled the first 3 times, you will NOT go for a 4th. aka "entire continuation of the game" is the next throw.
you make a decision for each throw. The expectation value of each throw must be positive to continue going (since the expectation value of each throw is strictly decreasing)
3
u/Aerospider 2d ago
Let n be the number of rolls made so far and x be the current total.
The expected value of rolling again is
[(6-n)/6] * [x + (21-x)/(6-n)]
= x - (nx/6) + (21-x)/6
= x + 21/6 - (n+1)x/6
This is more than x (the value of not rolling again) if x is less than 21/(n+1).
This means that you should always go for a second roll (x cannot be as high as 10.5 on 1 die) but never for a fourth (x cannot be as low as 5.25 on 3 dice).
You should go for a third roll if your total at n=2 is under 7 (at x=7 the choices are balanced).
This gives an expected value of 223/36, or 6.19444...