I am struggling to find a solution to this real life situation: -Given 4 groups A B C D, with different probabilities of an outcome Z (pAz pBz pCz pDz) -given that the number of people in each group is unequal - the probability of the outcome in different people is independent what is the overall probability of the outcome Z (overall means among all groups, among all people)? If only one group was present, I could easily solve it with binomial distribution or combinations. Given that there are 4 groups with different probabilities and unequal number of participants, what should I use? Should I use multinomial distribution?
Edit: there is no process to select people in each group, this is are given data I'd like to know the overall possibility of outcome Z ie:
- Group A 4 people, having probability of Z=0.03 group B 3 people: probability of Z= 0.05 group C 3 people: probability of Z= 0.07 group D 4 people: probability of Z= 0.08
what is the probability of Z? I think the solution should account for all possible combinations (taken 2 by 2, 3 by 3 and so on). If there was only one group, i could simply sum the binomial distributions.
edit2: trying to explain better with an example. There is only 1 suite in the hotel. People
- from city A will want it in 3% of cases
- from city B in 5% of cases
- from city C in 7% of cases
- from city D in 8% of cases
Given that tomorrow are going to arrive 4 people from A, 3 from B, 3 from C, 4 from D. What chances are the suite is used?