Because there are 18 red and 20 non-red numbers on the wheel, it follows there are 18 + 20 = 38 total numbers on the wheel. Since the wheel is equally likely to land on any of the numbers, the probability of getting a red number is 18/38 and the probability of getting a non-red number is 20/38.
The formula for the expected value of any wager with mutually exclusive outcomes is found by multiplying the probability of each outcome by the payout for that outcome, and subsequently summing the products. In this case the payout is $1 for red and -$1 for a non-red number. Thus the expected payout is $1*18/38 - $1*20/38 = -1/19*$1 which is about negative 5.3 cents. Rounding to the nearest decimal place and expessing the answer in terms of dollars, we would get -$.053 which rounds to -$.1.