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Q. Which of the following cannot be valid assignment of probabilities for outcomes of sample space
$S = \left\{\omega_{1}, \omega _{2}, \omega _{3}, \omega _{4}, \omega _{5}, \omega _{6}, \omega _{7}\right\}$ ?
Assignment
$\omega_{1}$ $\omega_{2}$ $\omega_{3}$ $\omega_{4}$ $\omega_{5}$ $\omega_{6}$ $\omega_{7}$
(i)$\,\,\,\,$ $0.1\,\,\,\,$ $0.01\,\,\,\,$ $0.05\,\,\,\,$ $0.03\,\,\,\,$ $0.01\,\,\,\,$ $0.2\,\,\,\,$ $0.6\,\,\,\,$
(ii)$\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$ $\frac{1 }{ 7}\,\,\,\,$
(iii)$\,\,\,\,$ $0.1\,\,\,\,$ $0.2\,\,\,\,$ $0.3\,\,\,\,$ $0.4\,\,\,\,$ $0.5\,\,\,\,$ $0.6\,\,$ $0.7\,\,$
(iv)$\,\,\,\,$ $-0.1\,\,\,\,$ $-0.2\,\,\,\,$ $-0.3\,\,\,\,$ $-0.4\,\,\,\,$ $-0.2\,\,\,\,$ $-0.1\,\,\,\,$ $-0.3\,\,\,\,$
(v)$\,\,\,\,$ $\frac{1 }{14}\,\,\,\,$ $\frac{2 }{14}\,\,\,\,$ $\frac{3 }{14}\,\,\,\,$ $\frac{4}{14}\,\,\,\,$ $\frac{5 }{14}\,\,\,\,$ $\frac{6 }{14}\,\,\,\,$ $\frac{15 }{14}\,\,\,\,$

Probability

Solution:

(i) Sum of all probabilities
$= 0.1 + 0.01 + 0 .0 5 + 0.03 + 0 01 + 0 .2 + 0 .6 = 1$
Hence, the assignment is valid, as sum is $1$
(ii) Valid, as sum is $1$ (sam e as part $(i)$),
(iii) Sum of all probabilities
$= 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0 .6 + 0 .7 = 2.8$
Hence, assignment is not valid, as sum is $2.8 > 1$.
(iv) Not valid, as probabilities of any event can't be negative.
(v) Not valid, as $ \frac{18}{7} > 1$, which is not possible.