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.