The probability that at least one condition is suffered on a day is 1 minus the probability that no conditions are suffered. That latter is simply the product of the probabilities of not suffering each condition.
Using the example probabilities of 3/7 for suffering the two conditions, we get 1 - (4/7)*(4/7) = 1 - 16/49 = 33/49.
This agrees with the answer given, but you should be able to see how easy it would be to extend. For instance, suppose three conditions with occurrences of 2/7, 2/7 and 3/7. The probability of any occurrence is 1 - (5/7)*(5/7)*(4/7) = 1 - 100/343 = 243/343, which is about 71%. Etc.
