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.