Two numbers are chosen independently and uniformly at random from the set {1,…
2019
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 ___________.
Attempted by 80 students.
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Correct answer: 0.5 to 0.504
Key idea: the most significant bit (MSB) in a 4-bit unsigned binary number is the 8's place, so it is 0 for numbers 1–7 and 1 for numbers 8–13.
Count how many numbers have each MSB value: 1 through 7 have MSB = 0 (7 numbers); 8 through 13 have MSB = 1 (6 numbers).
Probability both chosen numbers have MSB = 0 is (7/13)^2. Probability both have MSB = 1 is (6/13)^2.
Total probability = (7/13)^2 + (6/13)^2 = 85/169 ≈ 0.502958...
Rounded to three decimal places, the probability is 0.503.