Multiple choice

A coin of diameter $\dfrac {1}{2}$ units is tossed randomly onto the rectangular cartesian plane, the probability that the coin does not intersect any line whose equation is of the form $x = k, k \epsilon I$ is

  1. $\dfrac {1}{\sqrt 2}$
  2. $1-\dfrac {1}{\sqrt 5}$
  3. $\dfrac {1}{4}$
  4. $\dfrac {1}{2}$
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D Correct answer
Explanation

The coin has diameter 1/2, so its radius is 1/4. For the coin not to intersect any line x = k, its center must be at a distance greater than 1/4 from any integer line. This means the center must lie in the interval (n + 1/4, n + 3/4) for any integer n. The length of this interval is 1/2, and the total distance between integer lines is 1, so the probability is 1/2.

AI explanation

The integer grid lines x = k are spaced exactly 1 unit apart. The coin has a diameter of 1/2, so its radius is 1/4, and it avoids intersecting the vertical lines only if its center lands more than 1/4 unit away from any integer coordinate. Within each 1-unit wide strip, the safe region for the center has a width of 1 minus 2 times 1/4, which leaves a width of 1/2. The probability is the safe width of 1/2 divided by the total strip width of 1. Therefore, the result is 1/2.

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