Find the associated weights for each outcome.

Question

involving weights of outcomes in a sample space S = { O1, O2, O3, O4 }. Assume that Pr[ { O1, O2 } ] = 0.51, Pr[ { O1, O3 } ] = 0.56, and Pr[ { O1, O2, O3 } ] = 0.77. Find the associated weights for each outcome.
(1) w1 =

(2) w2 =

(3) w3 =

(4) w4 =

Answer 1

The most important idea is that the sum of the probabilities for all outcomes must equal exactly 100% or 1.00 as a decimal.

 

Now, O1 + O2 = 0.51   (Eq. 1)   while O1 + O3 = 0.56   (Eq. 2)    and      O1 + O2 + O3 = 0.77     (Eq. 3)

 

By examing the first equation together with the third equation, one may conclude that O3 = 0.26 by itself since 0.77 is greater than 0.51 by exactly 0.26. In a similar way, we may conclude that O2 must be 0.21 when we examine equations (2) and (3), We may then conclude that O1 must be 0.30 by examing any one of the 3 equations and by substituting the known values. Also, O4 must be 0.23

 

w1= 0.30
w2 = 0.21

w3= 0.26

w4= 0.23

 

If you add these four together, you get 1.00 as required.