Assume we start with 1.
In the first bet the expected value of remained balance is 1.5 * 0.5 + 0.5 * 0.5 = 1 For N times, is it still 1 according to E(XYZ)=E(X)E(Y)E(Z)? But 1.5^50 * 0.5^50 is not 1.
If the game is repeated N times, what's the expected value of remained balance in the end?
Thanks in advance!