Ball and drawer principle (aka modified pigeonhole)

Level pending

Consider a drawer full of balls of two colour - Black and White. Alice randomly selects one ball and takes it out. Then the same way she randomly selects another ball.

Given that the number \( n \) of black balls satisfies the condition that the probability that the two balls Alice chose are White is \( \frac{1}{2} \), and such that \( n \) is a \( 7 \)-digit number, find \( n \pmod {1000} \).

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