Multiple choice

From each of two equal lines (of length $i$) a portion is cut off at random and removed. What is the chance that the sum of the remainders is less than $i$ ?

  1. $\displaystyle \dfrac{1}{3}$
  2. $\displaystyle \dfrac{1}{2}$
  3. $\displaystyle \dfrac{2}{3}$
  4. $\displaystyle \dfrac{1}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let x and y be the lengths cut from two lines of length i. x, y are independent variables uniform on [0, i]. We want P( (i-x) + (i-y) < i ) = P( 2i - x - y < i ) = P( x + y > i ). The region x+y > i in the square [0, i]x[0, i] has area 1/2 of the total area.

AI explanation

Let the lengths cut off be x and y, where both are between 0 and i; the sample space is a square of area i squared. We need the sum of the remainders to be less than i, setting up the inequality (i minus x) plus (i minus y) is less than i, which simplifies to x plus y is greater than i. This inequality represents the area of the square above the line x plus y equals i, which forms a right triangle with half the area of the square. The probability is exactly 1/2.

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