Question

Which of the following cannot be valid assignment of probabilities for outcomes of sample space
S = $$\left\{\begin{array}{ccccccc}{w}_1,& w_2,& w_3,& w_4,& w_5,& w_6,& w_7,\end{array}\right\}$$

$$\begin{array}{cccccccc}& w_1& w_2& w_3& w_4& w_5& w_6& w_7\\ \left(a\right)& 0.1& 0.1& 0.05& 0.03& 0.01& 0.2& 0.6\\ \left(b\right)& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}\\ \left(c\right)& 0.1& 0.2& 0.3& 0.4& 0.5& 0.6& 0.7\\ \left(d\right)& -0.1& 0.2& 0.3& 0.4& -0.2& 0.1& 0.3\\ \left(e\right)& \frac{1}{14}& \frac{2}{14}& \frac{3}{14}& \frac{4}{14}& \frac{5}{14}& \frac{6}{14}& \frac{15}{14}\end{array}$$

Answer 1

Here probability of each outcome is positive and less than 1 and sum of probabilities is
= 0.1+ 0.01 + 0.05 + 0.03 + 0.01 + 0.2 + 0.6 = 1
$\therefore$ Both the conditions of axiomatic approach are satisfied.
Thus the assignment is valid.
(b) Here probability of each outcome is positive and less than 1 and sum of the probabilities is
$=\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}=1$
$\therefore$ Both the conditions of axiomatic approach are satisfied.
Thus the assignment is valid.
(c) Here probability of each outcome is positive and less than 1 and sum of the probabilities is
= 0.1 + 0.2 + 0. + 0.4 + 0.5 + 0.6 +0.7 = 2.8 > 1
$\therefore$ One of the conditions of axiomatic approach is not satisfied
Thus the assignment is not valid.
(d) Here probabilities of two events $w_1$ and $w_5$ are negative.
(e) Here probability of event $w_7$ is greater than 1. thus the assignment is not valid.