TABLE OF CONTENTS The binomial expansion is given by
or
where n is the total number of units in a system. As we know that R + Q = 1, then also ( R + Q)n = 1. Each term in the expanded form of ( R + Q) n may also be obtained from the following pdf of the discrete binomial distribution for k failures:
which gives the probability of exactly k failures. The cumulative binomial function is given by
Let's take a system with three units in it; then n = 3 and
( R + Q) 3 = R 3 + 3 R 2 Q + 3 RQ 2 + Q 3 = 1.
This says that the sum of the probabilities of all outcomes in a mission of length t, for which the R's and Q's are calculated, is equal to 1, as it should be.
What are these outcomes and their probabilities? The binomial distribution gives these to us very conveniently. The first term, R 3, gives the probability of all three units succeeding, or surviving, the mission. The second term without the coefficient, R 2 Q, gives the probability of two out of the three units surviving and the third failing, or RRQ. With the coefficient, it is 3 R 2 Q because all probabilities of two units surviving and one...
