Hi

Well, I have a problem here I need help to solve. Probability theory is not my field.

If someone knows how to solve it or knows where I can find tools for it, please contact me or just reply this thread as you please.

Take the definition:

Let A_1,A_2,…, and A_k be elementary events with probability P_1,P_2,…,P_k, respectively.

We say t is an average number of tries to occur any desired event about A_1,A_2,…, and A_k iff t is the minimum number that the probability P(t,P_1,P_2,…,P_k) of the desired event happen in t tries is >=1-(1-(P))^(1/P), where P=min{P_1,P_2,…,P_k}.

Notes: 1-(1-x)^1/x is a decrescent function, so 1-(1-(1/P))^(P), where P=min{P_1,P_2,…,P_k}, is maximum.

1-(1-n)^1/x is a crescent function for any fixed value of n.

Given fixed values of P_1,P_2,…,P_k, one can calculate numerically the average time of any event, however, when these values aren´t fixed things may become more complicated when we try to stablish some general formula (or an lower and upper bound) for the average time t.

For example:

We can prove that the average number of tries to occur A_1 with probability P_1 is 1/P_1. (this is the trivial case that satisfies the definition above)

We can prove the average number of tries t to occur A_1 or A_2, respectively, with probability P_1 and P_2 is >=ln((1+(1-1/2^|P|)^(2^|P|)))/(ln((1-1/(2^|P|)))) (which is a lower bound) and =< 1/(P_1 + P_2) (which is a upper bound), where P=min{P_1,P_2}

Following that right spirit, I do the question:

What is the average number of tries (or an non trivial upper bound and non trivial lower bound) to occur the subsequence A_1,A_2,…, and A_k, in that right ordering and non necessarilly one next to another, each elementary event with probabilities P_1,P_2,…,P_k, respectively. Note that k, P_1,P_2,… and P_k are free variables.

The answer will likely come as t being limited by an upper and lower bound to the desired average number of tries, where these bounds are functions from (P,P´,k) to a natural number (where P=min{P_1,P_2,… and P_k} and P´=max{P_1,P_2,… and P_k}).

Obviously, trying to find the exact function that gives our average time could be herculean and worthless to what we desire. The result comes from a combination and the combination formulas change for different values of k.

Just a non trivial lower bound would be quite good.

Thank you very much.