Two numbers are chose independently


#1

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 ______


#2

4-bit unsigned binary representation of numbers belongs to set {1, … , 13}:

1: 0 0 0 1
2: 0 0 1 0
3: 0 0 1 1
4: 0 1 0 0
5: 0 1 0 1
6: 0 1 1 0
7: 0 1 1 1
8: 1 0 0 0
9: 1 0 0 1
10: 1 0 1 0
11: 1 0 1 1
12: 1 1 0 0
13: 1 1 0 1

Numbers 1,2,3,4,5,6,7 have same MSB i.e. 0
Numbers 8,9,10,11,12,13 have same MSB i.e. 1
So, Probability that Chosen two numbers have MSB 0 is: (7/13)x(7/13)
Probability that Chosen two numbers have MSB 1 is: (6/13)x(6/13)
So, Total Probability = (7/13)x(7/13) + (6/13)x(6/13)
= 85/169
= 0.5029
= 0.503 (Rounded off to 3 Decimal Place)