The number occurrences in each case is (approximately) a Poisson random variable with expected numbers of occurrences of 3 and 4.5 respectively per month.
For wrongly reporting H we assume that the true expected number of occurrences per month is 3 and use the Poisson distribution to calculate the probability p(5+) that in any given month 5 or more occurrences are reported.
Now the number of months in our six month window for which 5 or more occurrences are reported is a Binomially distributed random variable ~B(6,p(5+)). So the probability of wrongly reporting H is:
P = b(3;6,p(5+)) + b(4;6,p(5+)) + b(5;6,p(5+)) + b(6;6,p(5+))