I have changed this question a bit to make this more interesting, but if I can answer this then I can answer my actual one!
I have a net that I want to capture a chicken with. The net is circular and is a fixed radius R=120cm. When the net is thrown, it may not land perfectly centred. I have a set of probability values that the net will land on the chicken (Event A) within specific radii, e.g. P(net land within r=15cm)=0.1, P(net land r=45cm)=0.4 etc. I have 15 values here in total.
I also have a probability of success (Event B) given that Event A has occurred for each of the 15 values (sometimes the chicken can escape!). The feature of this is that the further from the centre the net lands, the higher the probability of the chicken escaping.
My goal is to calculate/estimate the probability of catching the chicken, but I am stumped how to combine all the values since they are not identical trials (so I don't think it can be a binomial probability).
After that, I need to expand it to say if I have 5 attempts, what is the probability of capturing the chicken. The difference here is that the 15 Event A probabilities change with each attempt (the Event B ones do not).
For the first attempt, you can calculate the probability of success
where the radius r is sum over the support.
The question is if the probabilities change in the second attempt,
are both attempts independent from each other?
If your set up is "once success, you will stop", then the probability that
you will stop in the 1st or 2nd attempt is
where the later part is by independence property.
It seems to me that the possibility for P(B) > 1 exists this way, I was expecting a 1 - sigma solution.
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