Suppose you play a game where you role a single dice and

Good question, +1. I believe the expected value of the game to be $4.67. Here's my thought process:

The expected value of any given roll is 3.5. However, when deciding whether or not to forsake the first roll, you need to determine the expected value of playing the game with two rolls (since you have a 2nd and 3rd roll option) NOT just one roll. So let's isolate the game to two rolls.

Expected Value of Game with 1 roll: 3.5.
Expected Value of Game with 2 rolls: (1/2)(5) + (1/2)(3.5) = 4.25 (you keep the first roll if its 4, 5, or 6)

Now let's consider a game with three rolls:

You only keep the first roll if its 5 or 6. Otherwise, you dump it, since the expected value of a game isolated to two rolls is 4.25.

Therefore, EV = (1/3)(5.5) + (2/3)(4.25) = 14/3 = 4.67.

Edit: Just realized Bunuel used almost this exact same logic, lol. Good answer.

Hi! I have two questions in caps as below. Please explain.

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=4.25
PLEASE EXPLAIN HOW AND WHY DID WE MULTIPLY 1/2X3.5

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$
SIMILARLY HERE WHY 2/3*4.25. I AM JUST UANBLE TO GET IT. PLEASE HELP

As we all know this forum is intended solely for GMAT aspirants. Unless I am totally ignorant, these sort of questions usually are not presented in Official GMAT exam. Please do not dilute the utility and value of this forum by posting such questions. I leave it for the moderator and others to decide.

I would suggest to keep these questions in some other miscellaneous category, like beyond scope of GMAT.
Really, I would prefer to put my effort on real GMAT type of questions.