SOLUTION: Suppose it costs $5.00 to roll a pair of dice. If you get paid the positive difference of the two numbers in dollars that appear on the dice. What is the expected value of the ga

Question 1184637: Suppose it costs $5.00 to roll a pair of dice. If you get paid the positive
difference of the two numbers in dollars that appear on the dice. What is
the expected value of the game? Is it a fair game?

Please write down detailed working steps of this question.
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!

Here are the 36 dice rolls with 2 dice:

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
 
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
 
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 
 
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 
 
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
 
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Here are the positive differences of those above:

  0     1     2     3     4     5
  1     0     1     2     3     4
  2     1     0     1     2     3
  3     2     1     0     1     2
  4     3     2     1     0     1
  5     4     3     2     1     0

$X        P(X)           E(X) 
------------------------------
$0        6/36              0
$1       10/36           10/36
$2        8/36           16/36 
$3        6/36           18/36
$4        4/36           16/36
$5        2/36            2/36
------------------------------
     total expectation = 62/36 = 31/18 = $1.72222... 
                         
No. it isn't fair, because the expected value of the
game is $1.72222... minus $5.00 = -$3.27777...

So you expect to average losing about $3.28 per game
if you played the game many times.

Edwin

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