What is the probability exactly 4 of 6 patients are injured as a result from a fall?

Question

Assume the probability any patient injured from a fall is 0.4.

Answer 1

Let p=0.4 then 0.6 is the probability of non-injurious falls. The binomial expansion gives us the exact probabilities of combinations of circumstances: (0.4+0.6)^n=1, representing the total probability=1 (certainty) of the sum of all possible circumstances. When n=6 it expands to 0.4^6+6*0.6*0.4^5+15*0.6^2*0.4^4+... The third term's coefficient 15 is used for 4 out of 6 and 2 out of 6 falls, because in the expansion, the coefficients go: 1 6 15 20 15 6 1. 15*0.4^4*0.6^2 is the probability of exactly 4 injurious falls and 2 non-injurious falls=0.1382=13.82%.