If I have a bag containing a number of marbles (n) and a certain
number of them are red (r), how can I calculate the expected number of
selections I must make from the bag in order to select a first red
marble? Marbles are not replaced after each selection.

I have observed the following relation: 

Expected selections = (n+1)/(r+1)

I came up with this formula by simply staring at a few sample cases
for long enough. I have found that I do not know how to prove this
formula for the general case. A satisfactory answer to this question
would provide a detailed and easy to follow explanation of a proof or
disproof for this empirically tested formula.

Cheers,
Nigel.

Hi nigglefish!
First of all, let me tell you that you got the correct answer. I found
the formula for the expected number of trials before the first success
at the following link:

CRYSTOOL 4.1
http://porter.llnl.gov/crystool4.1/crystool_theory.html

Although it's for a different application, it can very easily be
adapted to your case, and the mathemtical proof is there. Notice (I
didn't realized this at first) that the expressions in that page that
seem to be fractions between parentheses are actually combinatorials.
For some reason, they put a line between the 2 numbers that define the
combinatorial.

For more information on the hypergeometric distribution (which is need
for the proof), you may want to follow this link:

Hypergeometric Distribution
http://mathworld.wolfram.com/HypergeometricDistribution.html


Google search strategy
://www.google.com/search?hl=en&lr=&q=%22without+replacement%22+%22first+success%22&btnG=Search


I hope this helps!
Best wishes!
elmarto

Can you expand on what you mean by "selection" and "a first red marble" please?