In box we have $n$ black and $m$ white balls without replacement. Let's denote

$B_k$ - number of black balls drawed in first $k$ draws

$W_k$ - same for white ones

Let's assume that we drawed three black balls at the beggining ($B_3=3, W_3=0$). Compute

$\mathbb{P} \left( B_k-W_k \ge 2 \text{ for all k } \in \{ 2,3,\ldots, n+m \} \right)$


This is the same as asking the following question: suppose we have $n-2$ black balls and $m$ white balls, drawn without replacement. What is the probability that we have never drawn more white balls than black. Investigate ballot numbers (see for example here).


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