Gate2019 cs Q32

0. 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 – Numerical Type question

Answer
Comment
Array

Option : 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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