Question

Which of the following can not be valid assignment of probabilities for outcomes of sample space S = Assignment ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 (a) 0.1 0.01 0.05 0.03 0.01 0.2 0.6 (b) (c) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (d) –0.1 0.2 0.3 0.4 –0.2 0.1 0.3 (e)

Answer 1

The sample space S is given by, S={ ω 1 ,ω 2 ,ω 3 ,ω 4 ,ω 5 ,ω 6 }

(i)

ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7
0.1	0.01	0.05	0.03	0.01	0.2	0.6

Here, each of the numbers p( ω i ) are positive and less than 1.

Also, the sum of all probabilities,

P=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.

(ii)

ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7
1 7	1 7	1 7	1 7	1 7	1 7	1 7

Here, each of the numbers p( ω i ) are positive and less than 1.

Also, the sum of all probabilities,

P=p( ω 1 )+p( ω 2 )+p( ω 3 )+p( ω 4 )+p( ω 5 )+p( ω 6 )+p( ω 7 ) = 1 7 + 1 7 + 1 7 + 1 7 + 1 7 + 1 7 + 1 7 =1

Thus, the assignment is valid.

(iii)

ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7
0.1	0.1	0.3	0.4	0.5	0.6	0.7

Here, each of the numbers p( ω i ) are positive and less than 1.

Also, the sum of all probabilities,

P=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

The sum of all the probabilities cannot be greater than 1.

Thus, the assignment is invalid.

(iv)

ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7
−0.1	0.2	0.3	0.4	−0.2	0.1	0.3

Here, p( ω 1 ) and p( ω 5 ) are negative.

Thus, the assignment is invalid.

(v)

ω 1	ω 2	ω 3	ω 4	ω 5	ω 6	ω 7
1 14	2 14	3 14	4 14	5 14	6 14	15 14

Here, p( ω 7 )is greater than 1.

Thus, the assignment is invalid.