Dice Game and Expected Value

Date: 04/18/2005 at 13:11:20
From: ayesha
Subject: confusing dice


You pay $6 to play a game where you roll a die with a payoff as 
follows: $8 for a 6, $7 for a 5, and $4 for any other result.  What
are your expected winnings?  Is the game fair?

Date: 04/20/2005 at 09:40:36
From: Doctor Wilko
Subject: Re: confusing dice

Hi Ayesha,

Thanks for writing to Dr. Math!

First, I'll explain the problem the way a text book might, and then
I'll try to give you some insight into how to interpret the answer.

Here are the facts of the problem as you described it:

  Pay:  $6 to play

  Possible rolls:             1    2    3    4    5    6
  Probability of each roll:  1/6  1/6  1/6  1/6  1/6  1/6
  Winnings for each roll ($)  4    4    4    4    7    8  

I'll use that information to calculate the expected value of the game:

  4*[$4(1/6)] + $7(1/6) + $8(1/6) - $6  

  = -$0.83...

I multiplied the 'winnings' of each roll by the probability of that
roll (note that since there are four rolls that pay $4, I multiplied
that result times 4).  I added up all six outcomes and then subtracted
out the $6 I paid to play.

Because the expected value is a negative answer, we'd say that the 
game is NOT fair.  In the long run, players of this game will lose 
money.  A fair game would yield an expected value of zero.

So, now that we've calculated expected value and we know it's 
negative, how do we interpret this?

The expected value of this problem could be interpreted as:  If you 
were to play the game one time each week over a six-week period (and 
each time get a different outcome (1-6)), your _average_ loss would be 
$0.83 per play.  

Think of it like this:

If you play six times, and you get each result once, then you'll 
have paid 6 * $6 = $36 dollars to play. Likewise you'll have won

  $8 + $7 + $4 + $4 + $4 + $4 = $31

dollars back.  This means that on average, every _six times_ you play, 
you'll lose 5 dollars.  

So you'll _average_ a loss of 5/6 of a dollar (-$0.83) _each_ time you 
play.

Does this help?  Please write back if you have further questions.

- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/