50 cards having one blue side and one red side are placed on a table all with the red face up. After 50 random card flips (a flipped card now displays its blue side, and if flipped again, it displays its red side etc.), what is the expected number of BLUE faces showing?

I haven't ever studied any statistics (I will be next semester), but I find it very interesting. I was wondering if anyone could show me what the standard way of solving this would be?

I tried to solve it just using logic, but came across some problems:

The total unconstrained number of possible combinations of red and blue cards (before the 50 random flips constraint) is 2^50. I can see that any combination that includes an odd number of blue faces (and therefore an odd number of red faces) cannot happen as 50 is an even number of flips, e.g., if you flipped 49 different cards, you would have 49 blue faces and 1 red face with one flip left over; regardless of which card is flipped in the 50th flip, there will be an even number of blue faces displayed.

Therefore, the number of possible outcomes is $\displaystyle 2^{50} - ($$\displaystyle 50\choose{1}$$\displaystyle +$$\displaystyle 50\choose{3}$$\displaystyle +$$\displaystyle 50\choose{5}$$\displaystyle +...+$$\displaystyle 50\choose{49}$$\displaystyle ) = \frac{2^{50}}{2}=2^{49}$

So, there are $\displaystyle 50\choose{2}$$\displaystyle +$$\displaystyle 50\choose{4}$$\displaystyle +...+$$\displaystyle 50\choose{50}$ blue cards shown in the possible combinations, which is slightly less than half. So I guess slightly less than 25 blue cards on average?

Can anyone do this properly so I can see?

Thanks.