The probability that three bites won’t become infected is the probability that the first two didn’t, times the probability that the third didn’t (i.e. (3/5)2 x 3/5 = (3/5)3).
I’m hoping that by now you’re seeing a pattern. The probability that n bites won’t become infected is (3/5)n; and when n = 7, the reported number of bites you have, the probability is (3/5)7.
(3/5)7 is approximately 2.8%, which is the probability of no infection if the probability of infection of each bite is independent and equal to 40%. Alas, this means that the probability of getting an infection is 1 - .028, or about 97%.
Fortunately for you, this logic applies only if the chance that one bite will become infected is independent of the chance that any of the others will become infected, which is certainly not the case. If the cat's mouth is pure, the chance that any will be infected is much much less than 40%, and the chance that none will be much higher than 3%. Conversely if the cat has a potty mouth, no math in the world will save you. You're doomed.