The probability of being a nonsmoker is 240 / 300 = 0.80. The probability of not suffering from allergies is 210 / 300 = 0.70. The probability of being a nonsmoker and not suffering from allergies is 185 / 300 = 0.62. Since the question asks for the probability of being either a nonsmoker or not suffering from allergies we have to take the probability of being a nonsmoker plus the probability of not suffering from allergies and subtract the probability of being both: 0.80 + 0.70 − 0.62 = 0.88.
Alternatively: 1 − P(Smoker & Allergies) = 1 − (35 / 300) = 88.3%.
