Problem Solving

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?

For example, if you could only roll one time, the expected value would be 1/6*(1+2+3+4+5+6) = 3.5. However, with three opportunities to roll higher, this value should increase.

OA : 4.67 $

any help is greatly appreciated.

newgmattest wrote:Suppose you play a gain 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?

For example, if you could only roll one time, the expected value would be 1/6*(1+2+3+4+5+6) = 3.5. However, with three opportunities to roll higher, this value should increase.

OA : 4.67 $

any help is greatly appreciated.

This is not a gmat oriented question, you can safely ignore this one..!!!

Thanks a lot. I do understand that it's not typical GMAT questions, but still it's quite good to understand how to solve these kind of questions from GMAT perspective.

Hey gmattest - I love the problem, although I agree with manpsingh that it's too labor intensive for the GMAT.

I'd solve it this way - you want to look at the probability multiplied by the expected value of each sequence. So you have:

1/6 chance of a 6 right away, and you'd stop rolling ---> 1/6 * 6 = $1 expected value
1/6 chance of a 5 right away, and you'd stop rolling here too (you can calculate it but the odds of a higher expected value by continuing to roll are bound to be lower than 5/6) = $5/6 expected value
1/6 chance of a 4 right away, and you can calculate it but I think this is the point at which you'd probably roll again (since there's a 50% chance of doing at least as well on the next one, and if you don't you get one more crack at it). So here you then have to say:

1/6 chance of a 4 * 1/6 chance of a 6 on the next roll = 1/6*1/6*6 = $1/6 expected value
1/6 chance of a 4 * 1/6 chance of a 5 on the next roll = 1/6*1/6*5 = $5/36 expected value

And so on for any of the "roll again" options (you'd definitely roll again on 1, 2, or 3). Then you'd add up all the expected values of each sequence and that's how you'd arrive at $4.67 (which makes sense as a final number since everything will be in sixths).


I hope that helps - like I said, this is pretty labor-intensive for the GMAT but the thought process is really helpful. We've always taught probability and combinatorics in the same lesson mainly because so much GMAT probability is based on the number of sequences/arrangements that are possible outcomes, so there's definitely some synergy between the two.