Question

Select numbers a and b between 0 and 1
independently and at random, and let c be their sum. Let A, B, and C be the results when a, b, and c, respectively, are rounded to the nearest integer. What is the probability that A + B = C?

Answer 1

If a and b are decimals between 0 and 1, then
A and B could be either 0 or 1
The prob of a rounded up to 1 is 1/2 and
the prob of a becoming 0 is 1/2

forming a table:
A B -----C
0 0 ----- 0 e.g. .2+.1 = .3 --> 0
0 0 ----- 1 e.g. .4 + .4 = .8 ---> 1 ...... FALSE
0 1 ----- 1 e.g. .4 + .9 = 1.3 ---> 1
1 0 ----- 1 e.g. .9+.4 = 1.3 --> 1
1 1 ----- 1 e.g. .6 + .6 = 1.2 ---> 1 ... FALSE
1 1 ----- 2 e.g. .9+.9 = 1.8 ---> 2

I see 4 cases out of the 6 where A+B = C
so prob = 4/6 = 2/3