Question

Consider the sample space S=​{ᴏ₁, ᴏ₂, ᴏ₃, ᴏ₄, ᴏ₅​}. Suppose that Pr (ᴏ₁) =0.17 and Pr (ᴏ₂) =0.47.

​(a)

Find the probability assignment for the probability space when ᴏ₃​, ᴏ₄​, and ᴏ₅ all have the same probability.

​(b)

Find the probability assignment for the probability space when Pr (ᴏ₅) =0.15 and ᴏ₅ has the same probability as ᴏ₄ and ᴏ₅ combined.

1. ​The probability assignment is Pr (ᴏ₁)=

         Pr (ᴏ₂)=

​                                                  Pr (ᴏ₃)=

​                                                  Pr (ᴏ₄)=

​                                                                Pr (ᴏ₅)=

Answer 1

Here,

space S=​{ᴏ₁, ᴏ₂, ᴏ₃, ᴏ₄, ᴏ₅​}.

Pr (ᴏ₁) =0.17

Pr (ᴏ₂) =0.47

(a)-> Since ᴏ₃​, ᴏ₄​, and ᴏ₅ all have the same probability,

Let us consider the probability be 'p'.

we know that the total probability of sample space is 1.

Pr(ᴏ₁) + Pr(ᴏ₂) + Pr(ᴏ₃) + Pr(ᴏ₄​) + Pr(ᴏ₅) = 1

0.17 + 0.47 + p + p + p = 1

3p = 1 - 0.64

p = 0.36/3 = 0.12

Hence the probability assignment is given by:

Pr (ᴏ₁) =0.17

Pr (ᴏ₂)= 0.47

Pr (ᴏ₃)= 0.12

Pr (ᴏ₄)= 0.12

Pr (ᴏ₅)= 0.12

(b)->

Here,

Pr (ᴏ₅) =0.15 and ᴏ₅ has the same probability as ᴏ₄ and ᴏ₅ combined. which means that Pr (ᴏ₄) + Pr (ᴏ₅) = 0.15. which implies that Pr (ᴏ₄) = 0

Let the  Pr (ᴏ₃) = q. Then,

we know that the total probability of sample space is 1.

Pr(ᴏ₁) + Pr(ᴏ₂) + Pr(ᴏ₃) + Pr(ᴏ₄​) + Pr(ᴏ₅) = 1

0.17 + 0.47 + q + 0.15 = 1

q = 1 - 0.79 = 0.21

Hence the probability assignment is given by:

Pr (ᴏ₁) =0.17

Pr (ᴏ₂)= 0.47

Pr (ᴏ₃)= q = 0.21

Pr (ᴏ₄)= 0

Pr (ᴏ₅)= 0.15