Is there a way to calculate the expected value of the maximum run lenght for an experiment with N trials?
To be more clear, If I flip a perfect coin (50% chance of heads, 50% chance of tails) say 10,000 times, is there a way to calculate the maximum number of times in a row I would expect to get tails?
Yes, using indicator random variables.
Let X[sub]i[/sub] be equal to 1 if there is a run of at least i tails, and 0 otherwise. Then the expected value of X[sub]i[/sub] is just the probability that there is a run of at least i tails. There are known formulas to calculate that probability; see for example here.
Then the maximum run length is exactly equal to X[sub]1[/sub]+X[sub]2[/sub]+…, and so the expected maximum run length is equal to the sum of the expected values of the X[sub]i[/sub]'s, i.e. the sum of the probabilities from the previous step.
The linked page has more information; in particular it states the value of the longest expected run, which might also be of interest to you. (But note that “longest expected run” and “expected maximum run length” aren’t the same thing!)
Thanks!
I’m sure once I look your refrence over I’ll be back with more questions 
Well, my ability to post may be limited over the Thanksgiving long weekend…just to warn you. Not that there aren’t several other posters qualified to answer your question, of course.
It’s been awhile since I did anything like this, but I’ll be on the boards over the weekend. Ask away.