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 $A + B = C?$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

Solution:

Represent the choices as a point $(a, b)$ in the unit square. Each of $a$ and $b$ rounds to $0$ if below $\tfrac12$ and to $1$ otherwise, while $c = a + b$ rounds based on $\tfrac12$ and $\tfrac32.$

The equation $A + B = C$ holds in these regions:

if $a + b \lt \tfrac12$ then $A = B = C = 0;$ if exactly one of $a, b$ is at least $\tfrac12$ and $a + b \lt \tfrac32$ then that variable rounds to $1 = C;$ and if $a + b \ge \tfrac32$ then $A = B = 1$ and $C = 2.$

These regions consist of two corner triangles of area $\tfrac18$ each and two central strips, with combined area $\tfrac34.$ Since the square has area $1,$ the probability is $\tfrac34.$

Thus, the correct answer is E.

2004 AMC 12A Problem 20

Solution:

Problem 20 in Other Years