I hope, you can help me with that:
Consider a cluster of n machines. Each has a probability p of failing in a given period of time T.
a) What is the probability of at least one machine failure during this period of time?
b) For $$0 \le k \le n$$, what is the probability $$p_k$$ (in terms of k, n and p) of exactly k machines failing during T?
c) Show that the probabilities from b) satisfy: $$p_1 + p_2 + ... + p_n = 1 - (1-p)^n$$
Thank you very much!
+23254 Assuming that all machines are functioning independently:
a) If the probability of failing is p, then the probability of working is (1 - p).
The probability that all will keep working is (1 - p)^n; thus, the probability that at least one will fail is
1 - (1 - p)^n
b) The probability that exactly k machines will fail is n nCr k ·p^k·(1 - p)^(n- k)
c) Part b gives the individual probabilities, their sum will be the sum found in part a.
