GATE Computer Science (CS) 2020 Solved Papers

Question : 52 of 65

Marks: +1, -0

For

{n}>2

, let

{a} ∈\{0,1\}^{n}

be a non-zero vector. Suppose that

{x}

is chosen uniformly at random form

(0,1)^{n}

. Then, the probability that

∑↙{i-1}↖{n} {ax}_{ {i}}

is an number is _______ .

Solution:

Let

n=3

0 ∊<0,1]^{3}

Let

a=[\table 1 ; 0 ; 1]

x=0

or 1

∑↙{i=1}↖{3} a_{1} x_{i}=a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}

=x_{1}=x_{3} \;\; ( ∵ a_{1}=1, a_{2}=0, a_{3}=1)

=0

or 1 or 1 or

2 .

Let

a=[\table 0 ; 0 ; 1]

∑↙{i=1}↖{3} a x_{i}=a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}

=x_{3} \;\; ( ∵ a_{1}=a_{2}=0)

=0

or 1
Let

a=[\table 0 ; 1 ; 0]

again for

a=[\table 1 ; 0 ; 0], ∑↙{i=1}↖{3} a_{i} x_{1}=0

or 1
Let

a=[\table 1 ; 1 ; 0]

∑↙{i=1}↖{3} a_{1} x_{i}=x_{1}+x_{2}=0,0,1

, or 2
Similarly for

a=[\table 0 ; 1 ; 1]

∑↙{i=1}↖{3} a x_{i}=0,1,1

or 2
Total number of cases

=2 × 3+4 × 3=18

Now odd number of cases

=9

P=\;{9}/{18}=0.5

Go to Question:

Prev Question

Next Question