How likely is someone to develop asthma as an adult?

It can happen. ..at anytime in your life.most likely due to allergens in the environment.
Yes. Although it was thought at one time that all asthma developed in childhood even by some prominent allergists and pulmonologists, we now know that adults do develop asthma which is however less likely to be allergic than in childhood asthma. The risk for a person in the general population to get adult asthma(no childhood asthma) is probably around 1% in my estimate.