Question

The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $0.357,$ or into $\sqrt3$ and $2.5-\sqrt3.$ Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3?$

$\frac14$

$\frac25$

$\frac12$

$\frac35$

$\frac34$

Answer: B

Concepts:continuous probability rounding interval length

Difficulty rating: 1980

Small Hint:

Let the first number be $x$ uniformly distributed on $[0,2.5]$

Big Hint:

Find the intervals on which $x$ and $2.5-x$ round to integers whose sum is $3$

Solution:

Let the first part be $x\in[0,2.5],$ so the second is $2.5-x.$ Ignoring endpoints of probability zero, the rounded values sum to $3$ exactly when $\begin{gathered} \frac12\lt x\lt1\\ \text{or}\\ \frac32\lt x\lt2. \end{gathered}$ These intervals have total length $1,$ out of a sample interval of length $2.5.$ The probability is therefore $\frac{1}{2.5}=\frac{2}{5}.$

Thus the correct answer is B.

1987 AMC 12 Problem 26

Answer: B

Difficulty rating: 1980

Small Hint:

Big Hint:

Solution:

Problem 26 in Other Years