23/6. One would choose to take the second roll when the first die comes up 1 or 2 since in this case the expected value of rolling again and paying 1 dollar (namely 2.5 dollars) is greater than what you currently have. It follows that the expected pay-off of the game is 2/3 * (3+4+5+6)/4 + 1/3 * 2.5 =23/6.