Question

Let a sample space be $S=\left\{{\omega }_1,{\omega }_2,...,{\omega }_6\right\}$. Which of the following assignments of probabilities to each outcome is/are valid?
Outcomes

	${\omega }_1$	${\omega }_2$	${\omega }_3$	${\omega }_4$	${\omega }_5$	${\omega }_6$
(i)	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$
(ii)	$1$	$0$	$0$	$0$	$0$	$0$
(iii)	$\frac{1}{8}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$-\frac{1}{4}$	$-\frac{1}{3}$
(iv)	$\frac{1}{12}$	$\frac{1}{12}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{3}{2}$
(v)	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$

• A. Only $(i)$
• B. Only $(ii)$
• C. Only $(v)$
• D. Both $(i)$ and $(ii)$

Answer 1

Answer: (D)

(i) Condition I Each of the number $P\left({\omega }_i\right)$ is positive and less than one.
Condition II : Sum of probabilities
$=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=1$
Therefore, the assignment is valid.
(ii) Condition I : Each of the number $P\left({\omega }_i\right)$ is either $0$ or $1$.
Condition II : Sum of the probabilities $=1+0+0+0+0+0=1$
Therefore, the assignment is valid.
(iii) Condition I : Two of the probabilities $P\left({\omega }_3\right)$ and $P\left({\omega }_6\right)$ are negative, the assignment is nor valid.
(iv ) Since, $P\left({\omega }_6\right)$ $=\frac{3}{2}>1$, the assignment is not valid.
(v) Since, sum of probabilities
$=0.1+0.2+0.3+0.4+0.5+0.6=2.1$.
the assignment is not valid.