SOLUTION: In a game, three standard dice are rolled and the number of odd values that appear is used to advance your gamepiece (for example, the roll 2-3-1 would advance your gamepiece two s

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Question 805273: In a game, three standard dice are rolled and the number of odd values that appear is used to advance your gamepiece (for example, the roll 2-3-1 would advance your gamepiece two spaces). Produce a probability distribution for this experiment.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
In a game, three standard dice are rolled and the number of odd values that appear is used to advance your gamepiece (for example, the roll 2-3-1 would advance your gamepiece two spaces). Produce a probability distribution for this experiment. 
There are 6󬝲 = 216 ways the three dice can fall.

The number of ways to get 3 evens is 3󫢫 = 27, in which you
would advance your gamepiece 0 spaces.

The number of ways to get 3 odds is also 3󫢫 = 27, in which you
would advance your gamepiece 3 spaces.

That accounts for 27+27 or 54 of the 216 rolls. That leaves 
216-54=162 possible rolls.

The remaining 162 rolls will consist of either 2 odds and 1 even,
or 2 evens and 1 odd.

There will be the same number of each, so there will be 

162/2 = 81 ways to roll 2 odds and 1 even, in which you would advance 
your gamepiece 2 spaces

and there will be 

162/2 = 81 ways to roll 2 evens and 1 odd, in which you would advance 
your gamepiece 1 spaces

So, 
the probability of moving your gamepiece 0 spaces is 27/216 = 1/8 = 0.125
the probability of moving your gamepiece 1 spaces is 81/216 = 3/8 = 0.375
the probability of moving your gamepiece 2 spaces is 81/216 = 3/8 = 0.375
the probability of moving your gamepiece 3 spaces is 81/216 = 1/8 = 0.125 


  X           P(X)       Expect-
[No. of      [Prob-      ation
spaces        abil-        =
moved]        ity]       X稰(X)
-------------------------------
  0          0.125          0
  1          0.375        0.375    
  2          0.375        0.75
  3          0.375        1.125
-------------------------------
             1.000        2.25

You weren't asked to make the third column.
So you can omit it.  However in other problems
involving expectation, you will have to do that.

The expectation of 2.25 means that if you roll
the three dice many times, you will average 
moving the gamepiece 2.25 spaces per roll.  [You 
will never move it 2.25 spaces, because you'll 
either move it 0,1,2, or 3 spaces.  However you
will average moving it 2.25 spaces per roll.

Edwin
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