Suppose you play a game where you role a single dice and whatever you roll you get that dollar amount. For example, if you roll a 1 you get $1. If you roll a 6 you get $6. If you are unhappy with the first roll, you can roll again. However, if you get lower the second time, you cannot take the first roll. If you are unhappy with the second roll you can roll a third and final time. Again, if you get lower on the third roll, you have to keep this roll and cannot take the first or second roll.
What is the expected value of this game?	Choose the correct choice:
A) 4250	B) 4500	C) 467	D) 512	E) 850
Expected value for one roll is 1/6*(1+2+3+4+5+6)= 
<gadget id="calculator">1/6*(1+2+3+4+5+6)</gadget>
<output>7/2 = around 3.5</output>
3.5$. This part is easy.   Let's proceed: if it's 3.5$, when will I agree to continue the game and roll the second time? Well obviously if the first roll will give me the value less than 3.5 or 1, 2, 3. In other cases if I get 4, 5 or 6, it won't be clever to continue as I already have from my first roll the value higher than the expected value of the second. So what is the expected value of two rolls:   1/6*6+1/6*5+1/6*4+1/2*3.5=2.5+1.75= 
<gadget id="calculator">2.5+1.75</gadget>
<output>4.25</output>
4.25   The same for the three rolls: if I know that expected value for two rolls is 4.25$, so I'll continue to roll if the first roll will give me the value less than 4.25 or 1,2,3,4. So for three rolls expected value must be:   1/6*6+1/6*5+2/3*4.25=1.833+2.833=~4.67$	467
<result>C</result>
Suppose you play a game where you role a single dice and whatever you roll you get that dollar amount. For example, if you roll a 1 you get $1. If you roll a 6 you get $6. If you are unhappy with the first roll, you can roll again. However, if you get lower the second time, you cannot take the first roll. If you are unhappy with the second roll you can roll a third and final time. Again, if you get lower on the third roll, you have to keep this roll and cannot take the first or second roll.
What is the expected value of this game?