here let net winnning is X.
below is pmf of X:
P(X=7-3=4) =P(rolling a 6)=1/6
P(X=1-3=-2)=P(rolling other number than 6)=5/6
therefore from above:
expected net winning E(X) =xP(x) =4*(1/6)-2*(5/6)= $ -1
E(X2)=x2P(x) =42*(1/6)+(-2)2*(5/6)= 6
therefore standard deviation =sqrt(E(X2)-E(X)2) =sqrt(6-12) =sqrt(5)=$ 2.236
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