Looks correct to me.
Hello all,
I was told that this is how to solve the following problem, but I just can't totally buy it and get my head around it. So if any of you can verify that this is correct, I would be indebted to you.
You've got a fair die, and I propose the following game. If you roll a 1 or a 2, I will give you 10 dollars (and the game ends). If you roll a 3 or a 4, you get another roll but the prize is discounted by d. That is, if you roll four 3's in a row, then roll a one, I give you 10d^4. If you roll a 5 or a 6, the game is over and you get nothing.
I want to find the expected value of this game, lets call this expected value G. The way I was told to solve is the following. There is a 1/3 chance you a 10 dollar prize, a 1/3 chance you get another roll (discounted) and a 1/3 chance you get nothing. Hence....
G=1/3(10)+1/3(dG)
solving for G we get...
10/(3-d)
Is this correct? Thank you kindly for any advice.
Sincerely,
Nick
Hello, salohcin!
I cranked this out . . . using my usual baby-talk.
You roll a fair die.
If you roll a 1 or a 2, you win $10 dollars.
If you roll a 5 or 6, you get nothing.
If you roll a 3 or a 4, you get another roll but the prize is discounted by
That is, if you roll four 3's in a row, then roll a one, you win dollars.
Find the expected value of this game.
We have these two facts:
. .
What about the third option?
Win in 2 rolls:
. . Roll a 3 or 4 on the first roll:
. . The prize is dollars.
. . Roll a 1 or 2 on the second roll:
Hence: .
Win in 3 rolls:
. . Roll a 3 or 4 on the first roll:
. . Roll a 3 or 4 on the second roll:
. . The prize is dollars.
. . Roll a 1 or 2 on the third roll:
Hence: .
. . and so on . . .
Hence: .
. . . . . .
The geometrtic series has the sum: .
Hence: .
Therefore: .