For a given number $n$, let us denote by $p_n$ the probability that when randomly selecting a pair of integers $k, m$ satisfying the conditions $0 \leq k \leq m \leq 2^n$ (the selection of each pair is equally probable), the number $\binom{m}{k}$ will be even. Calculate $\lim_{n \to \infty} p_n$.

Let's think step by step and output the final answer within \boxed{}.
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