In this part of the task, Machine $A$ is examined more closely. 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfills'' in a sample is assumed to be binomially distributed with $p = 0.3$. (1) Determine the probability of the event ``Fewer than 30 underfills occur''. (2) Determine the probability of the event ``At least 40 underfills occur''. (3) Give an event in the given context whose probability together with the probabilities from (2) and (3) sums to 1. (4) Give an event in the given context whose probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.
In this part of the task, Machine $A$ is examined more closely. 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfills'' in a sample is assumed to be binomially distributed with $p = 0.3$.
(1) Determine the probability of the event ``Fewer than 30 underfills occur''.
(2) Determine the probability of the event ``At least 40 underfills occur''.
(3) Give an event in the given context whose probability together with the probabilities from (2) and (3) sums to 1.
(4) Give an event in the given context whose probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.