Expected payoff in a dice game

Question

I was solving questions for finding expectation values, I enocuntered a question which is bugging me, the question says,

You are given a fair $6$ sided die and you roll it. You can either choose to keep your roll and receive the observed value in dollars. Alternatively, you are allowed to roll again, but if the sum of your two rolls is at least $7$ you receive nothing. If the sum of the two rolls is less than $7$, you receive the second observed value in dollars. Assuming optimal play, what is your expected payout?

My approach: I thought if my expected payout on the second roll is greater than my face value on the first roll, I should reroll, this is true only for a face value of $1$ (in which case the expected payoff on the second roll is $\frac{1+2+3+4+5}{6}=\$2.5$), for all other face values it's better not to reroll.

From this logic, the expected value comes out to be $\frac{2.5+2+3+4+5+6}{6}=\$3.75$, but apparently this is not the right ansswer.

Please point the fallacy in my understanding and if possible, the right approach and answer, the link to the question is here

Answer 1

Hint

You have taken that your payoff is only what you get from the second roll, but though the wording could be better, I believe they mean that if the sum of the first+second roll is $6$ or lower, you get the entire amount (not merely that of the second roll), else zero.

If you work it out that way, you will find that the optimal strategy would be to reroll if first roll is $1$ or $2$

Rework your attempt in this way

Total payoff in 6 rolls

$1st\quad\quad After\;2nd$
$ 06 | (2+3+4+5+6+0) =20$
$12| (3+4+5+6+0+0) =18$
$18| (4+5+6+0+0+0) =15$
$24| (5+6+0+0+0+0) =11$
$30| (6+0+0+0+0+0) =06$
$36| (0+0+0+0+0+0) =00$

Optimal earnings/game = $(20+18+18+24+30+36)/36 = 4.055$