Let "B" be the event "draw a black marble from a cup". Then
"2 \\ cups:P(BB)={1 \\over 3}\\cdot{1 \\over 3}=({1 \\over 3})^2"
"3 \\ cups:P(BBB)=({1 \\over 3})^3"
"..."
"N \\ cups: P(B\\ \\text{n times})=({1 \\over 3})^N"
"P(\\text{at least 1 white marble if I have N cups)}="
"=1-P(B\\ \\text{n times})=1-({1 \\over 3})^N"
"100\\ cups:P(\\text{at least 1 white marble if I have 100 cups)}="
"=1-({1 \\over 3})^{100}\\approx"
"\\approx0.99999999999999999999999999999999999999999999999806"
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