mathematical expectation

heng87

A player is invited by paying RM4.00,to make a random selection of 2 marbles,one by one and without replacement,from a box containing 6 red marbles,3 green marbles and 1 white marble.For each red marble selected,he will be paid RM1.00,each green marble RM2.00 and each white marble RM6.00.Calculate the expected gain or loss after a selection of 2 marbles.
If a payment of RMW is now given to each of the outcomes (R,R) and (G,G),the expected gain after a selection of 2 marbles is RM1.00.Calculate the value of W.(Red marble,green marble and white marble are denoted by R,G and W respectively.

peaceboy

6，3，1
P(RR) = 6/10 * 5/9 = 1/3 , RM 2
P(GG) = 3/10 * 2/9 = 1/15 , RM4
P(GR) = 3/10 * 6/9 = 1/5 , RM 3
P(RG) = 6/10 * 3/9 = 1/5 , RM 3
P(WR) = 1/10 * 6/9 = 1/15 , RM 7
P(RW) = 6/10*1/9 = 1/15 , RM 7
P(WG) = 1/10 * 3/9 = 1/30 , RM 9
P(GW) = 3/10*1/9 = 1/30 , RM 9

E(X) = 2*1/3 + 4*1/15 + 3*1/5 + 3*1/5 + 7*1/15 + 7*1/15 + 9*1/30 + 9*1/30
      = 2/3+4/15+6/5 + 14/15 + 18/30
       =33/9

expected gain = 33/9 - 4
                     = 1/3
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E(X) = W*1/3 + W*1/15 + 3*1/5 + 3*1/5 + 7*1/15 + 7*1/15 + 9*1/30 + 9*1/30
       =6w/15 +41/15
       =(6W+41)/15
expected gain = (6W+41)/15 - 4 =1
(6W+41)/15=5
6W+41=75
6W=34
W=34/6