+1537 Suppose we have a bag with $10$ slips of paper in it. Eight slips have a $3$ on them and the other two have an $8$ on them.
How many $8$'s do we have to add to make the expected value equal to $4$?
+129852 Let the number of "8" slips we need to add = N
So
3 ( 8 / (10 + N) ) + 8 ( 2+ N) / (10 + N ) = 4 simplify
[ 24 + 16 + 8N ] / (10 + N) = 4
[ 40 + 8N ] = 4 (10 + N)
40 + 8N = 40 + 4N
8N = 4N
(8- 4) N = 0
4N = 0
N = 0 (we don't need to add any !!! )
Proof :
Expected value = Outcome * Probability = 3 (8/10) + 8(2/10) = [ 24 + 16 ] / 10 = 40 / 10 = 4
