In this part of the task, Machine $B$ is examined more closely. In the manufacturer's specifications for Machine $B$, it states that 30\% underfills can be expected. The responsible machine operator has the suspicion that Machine $B$ works better than specified. With a sample of 200 bottles, he tests his suspicion. If at most 45 underfills occur in this sample, he assumes that the machine works better than specified.
Determine the probability with which the machine operator assumes that the machine works better, even though the machine actually produces underfills with a probability of $p = 0.3$.