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A box contains a very large number of balls, so that the probability of choosing a white or red (initially at equal numbers) remains at 1/2 as balls are chosen. Let X be the number of balls chosen at random until a red ball is chosen. Determine the cumulative probability distribution of X.

I've tried with the idea that this might be a geometric progression, so the cumulative function will have p*(1-p)^k for the kth term, and p = 0.5, and summed that from 1->x to give a cumulative distribution function of 1-2^(-n). However, I'm not sure this is correct...

Thanks!

Laweng
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1 Answers1

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Note first that the cdf $F_X(x)$ of $X$ is equal to $0$ if $x\lt 1$. Now suppose that $x\ge 1$. Let $\lfloor x\rfloor$ be the greatest integer which is $\le x$. We have $$F_X(x)=\Pr(X\le x)=\Pr(X\le \lfloor x\rfloor)=1-\Pr(X\gt \lfloor x\rfloor).$$ But the probability that $X\gt \lfloor x\rfloor$ is the probability of $\lfloor x\rfloor$ white balls in a row, which is $\frac{1}{2^{\lfloor x\rfloor}}$. Thus $$F_X(x)=1-\frac{1}{2^{\lfloor x\rfloor}}$$ if $x\ge 1$.

Remark: Your answer is along the right lines. However, the cdf $F_X(x)$ is defined for all real values of $x$.

André Nicolas
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