Let \(\Omega\) be a sample space. Let \(m(a) = 1=2, m(b) = 1=3 , m(c) = 1=6\). Find the probabilities for all eight subsets of \(\Omega\)
\(\Omega\) is the space of: \(m(a) = \frac{1}{2}\) \(m(b) = \frac{1}{3}\) \(m(c) = \frac{1}{6}\)
\(P(\theta) = m(\theta) = 0\) \(P(a) = m(a) = \frac{1}{2}\) \(P(b) = m(b) = \frac{1}{3}\) \(P(c) = m(c) = \frac{1}{6}\)
The probabilities for each subst are: \(P(a,b) = m(a) + m(b) = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}\)
\(P(a,c) = m(a) + m(c) = \frac{1}{2} + \frac{1}{6} = \frac{2}{3}\)
\(P(b,c) = m(b) + m(c) = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}\)
\(P(a,b,c) = m(a) + m(b) + m(c)= \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1\)
This is the probabilities for all subsets.