One strategy is to always bet "red". what is p(red)? define x=#red in 38 spins. what is the probability you'll break even by betting on red-only in 38 spins? that is, what is p(19 red in 38 spins)?

Either you get "red" or you don't.  That's binomial.  The probability of getting "red" is 1/2, same as the probability of getting "black."  

So we have a binomial probability distribution with n=38, p=1/2 and x = 19.

Using the binompdf( function built into my TI-83 calculator, I get

binompdf(38, 1/2, 19) = 0.129 (answer)