Patients recover in a ward after various surgical procedures. There is a probability of recovery for each day of stay in the ward which depends upon which procedure was carried out and which surgeon carried out the procedure.
For example there is a probability 0.6 that patient A having had a hip replacement, carried out by surgeon B, will have recovered on day 4 of their stay in the ward.
Is it correct to assume that the calculation of the probability of 6 out of 10 free beds on day 1 could be treated as a sum of Binomial trials? Assuming that the recovery probabilities are independent.
P(6 beds free on day one) = P(beds 1,2,3,4,5,6 free)*p(beds 7,8,9,10 not free) + p(beds 2,3,4,5,6,7 free)*p(beds 1,8,9,10 not free + ...
Where P(beds 1,2,3,4,5,6 free) = P(bed 1 free)*p(bed 2 free)p(bed 3 free) ...
Where the probability of bed 1 being free depends on which day of recovery the patient is on, the type of surgical procedure that was carried out and which surgeon carried it out, assuming that these probabilities are known.
Are there any other approaches to calculating these probabilistic? For example using a probability generating function? A distribution as an approximation?