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I have 100 balls, which are all initially yellow. Every minute, I randomly choose a ball and paint it red.

How many balls are expected to be red after 100 minutes? Note: I could pick up a ball that's already painted red.

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  • $\begingroup$ Perhaps consider the variables $X_i=\cases{1, & if the $i$th ball is painted red\cr 0, & otherwise }$. (The number of red balls (at the end) is then $\sum_{i=1}^{100} X_i$.) $\endgroup$ – David Mitra Oct 3 '15 at 12:41
  • $\begingroup$ This is related to the coupon collector's problem - see en.wikipedia.org/wiki/Coupon_collector%27s_problem $\endgroup$ – Marconius Oct 3 '15 at 12:44
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Consider a random variable $X_i$ such that $X_i = 1$ if the $i^{th}$ ball is red after 100 minutes. Note that $Pr(X_i = 1) = 1 - ({99 \over 100})^{100}$. Using linearity of expectation,

$$E(X) = \sum_{i=1}^{100} E(X_i) = 100\left(1 - \left({99 \over 100}\right)^{100}\right) \approx 63.4$$

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