GATE | GATE CS 2019 | Question 29

Two numbers are chosen independently and uniformly at random from the set {1, 2, …, 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is ___________.

Note: This was Numerical Type question.
(A) 0.5029
(B) 0.538
(C) 0.461
(D) 0.248


Answer: (A)

Explanation: The 4-bit binary representation of numbers (1, 2, 3, 4………13):

0  - 0000 
1  - 0001 
2  - 0010
3  - 0011
4  - 0100
5  - 0101
6  - 0110
7  - 0111
8  - 1000
9  - 1001
10 - 1010
11 - 1011
12 - 1100
13 - 1101

There 6 numbers which start with MSB as 1, and 7 numbers which start with MSB as 0.

Therefore, probability that their 4-bit binary representations have the same most significant bit is,

= P(MSB is 0) + P(MSB is 1)
= (7×7)/(13×13) + (6×6)/(13×13)
= (49+36)/169
= 85/169
= 0.5029 

Option (A) is correct.

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