2
$\begingroup$

An urn contains 10 black marbles and some white. We are going to draw one marble at random. Before the draw, one more marble has been added in the urn. Then we draw the marble and it is a black one. What is the probability that the extra marble was black?

  • 0
    Well, I am not sure: Assume we have w white marbles. Then before the addition of the extra marble, the probability of drawing a black one was 10/(w+10) and the probability of drawing a white was w/(w+10). After the addition, if the extra marble was black, the probability of drawing a black would be 11/(w+11) and the probability of drawing a white would be w/(w+11). If the extra marble was white instead, the probability of drawing a black would be 10/(w+1+10) and the probability of drawing a white would be (w+1)/(w+1+10). So, all in all, the probability that it was black is 11/21? – 2017-02-19
  • 0
    But how do we take into account the fact that a black marble was drawn? – 2017-02-19

1 Answers 1

1

Suppose that we started with $w$ white balls in the urn and let us assume that there is an equal probability that the new ball is either white or black. Then we apply Bayes' Theorem to deduce that the probability that the new ball was black is $$\frac {\frac {11}{11+w}\times \frac 12}{\frac {11}{11+w}\times \frac 12+\frac {10}{11+w}\times \frac 12}=\frac {11}{11+10}=\frac {11}{21}$$

We note that this is independent of $w$, so this is our answer regardless of the distribution on the possible $w$. It does, however, depend on the probability that the additional ball is black so some assumption on that is required.

  • 0
    Dear lulu, are you sure the second fraction in the denominator is 10/(10+w)? Or maybe 10/(11+w)? – 2017-02-19
  • 0
    Good catch. Typo. I will edit. – 2017-02-19
  • 1
    That term represents the probability that you draw black, conditioned on the additional ball being white. The whole point here is that all three denominators are $11+w$....that's why we can get rid of the apparent dependence on $w$. – 2017-02-19
  • 0
    I see! Thanks a lot!! – 2017-02-19