On 13/01/2011 02:51 PM, Invisible wrote:

> Specifically, if the probability of catching the diabolo is P, then the
> probability of *not* catching it must be 1-P. Apparently the probability
> of K successes followed by one failure is
>
> Prob K = P^K * (1-P)

If P is the probability of an event happening, than P^K is the 
probability of it happening K times, and 1-P is the probability of it 
not happening once. Hence the above formula.

I tried plotting it:

http://tinyurl.com/6bsfnol

As you'd expect, as the catch probability goes up, the probability of 
longer chain lengths increases sharply.

Paradoxically, as P becomes very close to 100%, 1-P obviously becomes 
extremely close to zero. In other words, /every/ chain length becomes 
very unlikely.

If we look not at chains of length K but chains of length /at least/ K, 
we get Prob K = P^K. Plotting this:

http://tinyurl.com/5wlkdmf

we discover that indeed, as P increases, the typical chain lengths 
increase very sharply. (If Wolfram had bothered to label their axies, I 
might even be able to tell you what the probability of 50 catches is. :-P )

> All of this of course assumes that the trails are *independent*, which
> they manifestly are not...

I have no idea how to account for that mathematically, however.

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