# Statistics puzzler

William Bains william at wbains.u-net.com
Fri Oct 31 04:24:20 EST 1997

```In article <34576CFD.115A at nrc.uab.edu>, rdudley at nrc.uab.edu wrote:

> You are given 50 black, and 50 white marbles.  Your job is to separate
> them into two urns so that there is the greatest chance of picking a
> white marble.
>
> Her answer, which is apparently the correct one, was to put only one
> white marble in on urn, and the rest of the marbles in the other urn,
> thus giving an approx. 75% chance of selecting a white marble (by adding
> the simple probabilities of 50% chance of selecting a 100% success, plus
> 50% chance of selecting a 50% success rate).
>
> That sounded fishy to me.  I argued that the greatest chance is still
> only 50%, because the urns represent two separate populations of
> marbles, comprising a 50% success rate, and a 25% success rate.
>

This sounds right. This is actually a two-stage selection procedure -
stage 1 is select an urn at random,. and stage 2 is select a marble at
random. So, if one urn (urn 1) is guarenteed to give to a white marble and
the other one (urn 2) is about 50:50, then the probability that you chose
a white marble is

0.5x(prob white marble in urn 1) + 0.5x(prob. white marble in urn 2)

= 0.5x1 + 0.5x0.5 = 0.75

Incidentally, this is probability, not statistics, I think.

William Bains

```