I've been struggling with this for a number of days now and so decided to ask this question. This may be trivial but I'm not particularly good or well-educated in statistics. So here goes nothing.
What I need to calculate is the overall probability of at least one out of a number, say N, events occurring. There is a number of things that I know about all the events:
- all events are disjoint, i.e. there isn't a chance that any two can occur in the same sample (
P(A or B) = P(A) + P(B))
- events are not independent, specifically if one occurs all the following ones cannot occur
- probability of any of the events occurring (computed externally, therefore not in the same sample space)
All this leads me to a conclusion that the probability of any of the events occurring should be the same as exactly one of them occurring, since that is the only possible outcome (out of a series of N events only 1 or 0 could happen).
What I've come up with so far is that for the first event out of the series the probability will be simply the probability of that particular event:
P(E_1) = P_1 // I know P_1 etc.
For the second event to happen event 1 must not have happened. So the probability of the second event is:
P(E_2) = (1-P_1)*P_2
I have doubts about this, because I know that:
P(E_2 and E_1 complement) = P(E_2) - P(E_2 and E_1) = P(E_2) // Since P(E_2 and E_1)=0 because E_2 and E_1 cannot happen simultaneously.
Ignoring my doubts, I've generalised this to the following for N events (probability of Nth event occurring):
P(E_N) = (1-P_1)*(1-P_2)*...*(1-P_N-1)*P_N
Now, since the events are disjoint, the probability of any of them occurring (which, given the conditions, should be the same as exactly one occurring) is:
P(any event out of N) = P(E_1 or E_2 or … or E_N) = P_1 + (1-P_1)*P_2 + (1-P_1)*(1-P_2)*P_3 + … + (1-P_1)x(1-P_2)*...*(1-P_N-1)*P_N
Question 1 Is the above correct? Or am I simply complicating things too much and the combined probability of exactly one event occurring will be simply the sum of the probabilities of all the events?
Question 2 Now say I've computed the probability of one out of N events occurring correctly. If I add another, N+1th event, will any changes need to be made to account for the fact that the sample space may have changed? I worry that as N tends to infinity the probability of any of the events occurring will exceed 1 which will be awkward as hell.
Thanks a lot for the help. And apologies for not formatting the maths bits nicely, I'm new to the Maths part of Stack Exchange.