Question

Which of the following can not be a valid assignment of probabilities for outcomes of sample Space $S =\\{ω_1, ω_2 , ω_3, ω_4, ω_5, ω_6, ω_7\\}$

Answer 1

(a)

Here, each of the numbers p(ωi) is positive and less than 1.
Sum of probabilities
$=p(ω_1)+p(ω_2)+p(ω_3)+p(ω_4)+p(ω_5)+p(ω_6)+p(ω_7)$
$=0.1+0.01+0.05+0.03+0.01+0.2+0.6$
$=1$

Thus, the assignment is valid.

(b)

Here, each of the numbers p(ω) is positive and less than 1.
Sum of probabilities
$=p(ω_1)+p(ω_2)+p(ω_3)+p(ω_4)+p(ω_5)+p(ω_6)+p(ω_7)$
$=\dfrac{1}{7}+\dfrac{1}{7}+\dfrac{1}{7}+\dfrac{1}{7}+\dfrac{1}{7}+\dfrac{1}{7}+\dfrac{1}{7}=7×\dfrac{1}{7}=1$
Thus, the assignment is valid.

(c)

Here, each of the numbers p(ωi) is positive and less than 1.
Sum of probabilities
$=p(ω-1)+p(ω_2)+p(ω_3)+p(ω_4)+p(ω_5)+p(ω_6)+p(ω_7)$
$=0.1+0.2+0.3+0.4+0.5+0.6+0.7$
$=2.8≠1$
Thus, the assignment is not valid.

(d)

Here, p(ω_1)$$$ and p(ω_5)$p(ω_5) are negative.
Hence, the assignment is not valid.

(e)

Here,$P(ω_7)=\dfrac{15}{14}>1$

Hence, the assignment is not valid.