Let \(A_{i}, i=1, \ldots, 4\), represent four events in a sample space, \(S\). For each of the situations below, determine which assignment of probabilities are actually possible (i.e., do not contradict Kolmogorov's axioms), and which are not. Justify your answers.

a. \(P\left(A_{1}ight)=.3, P\left(A_{2}ight)=.3, P\left(A_{3}ight)=.2, P\left(A_{4}ight)=.2\)

b. \(P\left(A_{1}ight)=.3, P\left(A_{j}ight) \geq P\left(A_{1}ight)\) for \(j=2,3,4\)

c. \(P\left(A_{1}ight)=.7, P\left(A_{2}ight)=.6, P\left(A_{1} \cap A_{2}ight)=.1\)

d. \(P\left(A_{1}ight)=.7, P\left(A_{2}ight)=.6, P\left(A_{1} \cup A_{2}ight)=.1\)

e. \(P\left(A_{1}ight)=.3, P\left(A_{2}ight)=.4, P\left(A_{3}ight)=.1, P\left(A_{4}ight)=.2\) where \(A_{i} \subset A_{j}, i

f. \(P\left(A_{1}ight)=.4, P\left(A_{2}ight)=.3, P\left(A_{1} \cup A_{2}ight)=.5\) where \(A_{1} \cap A_{2}=\emptyset\)