Grid & Crosses

Probability Level 3

As shown above, a grid, composed of multiple 1 × 1 1\times 1 square units, is constructed such that any distance of the vertical and horizontal white connecting squares equals 3 3 units, alternating with the black centers and crosses. This pattern continues and expands endlessly.

If one square is picked randomly from this grid, what is the absolute difference between the probability of obtaining a white unit and that of a black one?

If your answer is in a form of a b \dfrac{a}{b} for co-prime a , b a,b integers, enter a + b a+b as the solution.


The answer is 5.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

When analyzing this grid deliberately, we can see that it is created from multiple building blocks of congruent 4 × 4 4\times 4 squares unit, as depicted with the red margin, and in each of this larger unit, there are 6 6 black squares and 10 10 white squares.

Hence, with large n n units of these, the probability of obtaining a white square = lim ⁡ n → ∞ 10 n 16 n = 5 8 \displaystyle{\lim_{n \to \infty} \dfrac{10n}{16n}} = \dfrac{5}{8} .

Similarly, the probability of obtaining a black square = lim ⁡ n → ∞ 6 n 16 n = 3 8 \displaystyle{\lim_{n \to \infty} \dfrac{6n}{16n}} = \dfrac{3}{8} .

Thus, the difference between these probabilities = 5 8 − 3 8 = 2 8 = 1 4 \dfrac{5}{8} - \dfrac{3}{8} = \dfrac{2}{8} = \boxed{\dfrac{1}{4}} .

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