Re: Fun probability problem

Stephen Montgomery-Smith wrote:

Suppose you have 100 balls in an urn, and 15 of them are white. What is the expected number of balls you should pick randomly from the urn (without replacement) before you get a white ball?

Obviously one can generalize the problem where 100 is n and 15 is m.

I found one rather difficult way (compute the probability that the first white ball is picked the kth time, and then compute the sum). I also found a very slick way, but only after doing it the hard way.

This problem must be well known. Does anyone have a reference?


Label the eighty-five non-white balls 1 through 85 Let I_i be the indicator variable for the event that non-white ball i is drawn before any of the white balls. Then the number of balls drawn up to and including the first white ball is 1 + sum(i=1..85, I_i). Hence, its expected value is
1 + sum(i=1..85, EI_i) = 1 + 85 * 1/16 = 101/16. (Each of non-white ball i and the fifteen white balls is equally likely to come up first amongst the sixteen of them.)

Yes, this is well known. As others have noted, Ross considers the case of the number of cards to turn over before the first ace. See his chapter (in the First Course book) on Expectation. I like to call it the "Hypergeometric Process" (cf. a Bernoulli process), but I don't think I have seen anyone else use that terminology.

Exercise: Determine the variance.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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Relevant Pages

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    ... first white ball is picked the kth time, ... of the number of cards to turn over before the first ace. ... chapter on Expectation. ... elegant, neat, and correct. ...
    (sci.math)
  • Re: Fun probability problem
    ... I found one rather difficult way (compute the probability that the first white ball is picked the kth time, ... Ross considers the case of the number of cards to turn over before the first ace. ... See his chapter on Expectation. ... The problem of finding the variance has at least one solution which is elegant, neat, and correct. ...
    (sci.math)
  • Re: Fun probability problem
    ... I found one rather difficult way (compute the probability that the ... first white ball is picked the kth time, ... chapter on Expectation. ... The probability of picking b black balls without a white ball is ...
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