Which of the following can not be valid assignment of probabilities for outcomes of sample space
Question:

Which of the following can not be valid assignment of probabilities for outcomes of sample space $S=\left\{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}, \omega_{5}, \omega_{6}, \omega_{7}\right\}$

Solution:

Here, each of the numbers pi) is positive and less than 1.

Sum of probabilities

$=p\left(\omega_{1}\right)+p\left(\omega_{2}\right)+p\left(\omega_{3}\right)+p\left(\omega_{4}\right)+p\left(\omega_{5}\right)+p\left(\omega_{6}\right)+p\left(\omega_{7}\right)$

$=0.1+0.01+0.05+0.03+0.01+0.2+0.6$

$=1$

Thus, the assignment is valid.

Here, each of the numbers pi) is positive and less than 1.

Sum of probabilities

$=p\left(\omega_{1}\right)+p\left(\omega_{2}\right)+p\left(\omega_{3}\right)+p\left(\omega_{4}\right)+p\left(\omega_{5}\right)+p\left(\omega_{6}\right)+p\left(\omega_{7}\right)$

$=\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}=7 \times \frac{1}{7}=1$

Thus, the assignment is valid.

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Here, each of the numbers pi) is positive and less than 1.

Sum of probabilities

Here, $p\left(\omega_{1}\right)$ and $p\left(\omega_{5}\right)$ are negative.

Hence, the assignment is not valid.

Here, $p\left(\omega_{7}\right)=\frac{15}{14}>1$

Hence, the assignment is not valid.