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 ______

# Two numbers are chose independently

**arjya11**#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)