I have a problem, which looks like a regular probability one, but has much greater numbers, then usually met. I have a 500'000 balls in a bag, one ball is white and all others are black. We draw a ball, look at it and throw it back in the bag.
How many times should one draw a ball and throw it back until a white one is drawn with probability 95%?
The regular solution with degrees doesn't work here. The 499999/500000 is too small and the power is too high. How this can be addressed?
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Let's think of this as a geometric distribution. Let X be the random variable 'number of trials until a white ball is pulled'. Then $X \sim Geo(\frac{1}{500000})$. We want to search for an $n$ such that $P(X\leq n) \geq 0.95$.
$ \Rightarrow 1-(1-p)^n\geq0.95 $
$ \Rightarrow (1-p)^n\leq 0.05 $
$ \Rightarrow n \geq \log_{(1-p)}(0.05) = \log_{\frac{4999999}{500000}}(0.05) \approx 1.498\times10^{6} $
So you'd want to draw roughly 1.5 million balls to be 95% sure that you're going to have draw a white ball at some point.
A Poisson approximation seems appropriate given the small probability of drawing a white ball.
Let $X$ be the total number of white balls in $n$ draws. We assume $X$ follows a Poisson distribution with mean $$\lambda = n / 500,000$$ So $$P(X > 0) = 1 - \exp(-\lambda) = 1 - \exp(-n / 500,000)$$ We want $P(X>0) > 0.95$, i.e. $$1 - \exp(-n / 500,000) > 0.95 $$ so $$\begin {align} \exp(-n / 500,000) &< 0.05 \\ -n / 500,000 &< \ln(0.05) \\ n &> -500,000 \cdot \ln(.05) \approx 1.50 \times 10^6\\ \end{align}$$