Question

A device consists of 100 independent modules of equal functionality. Zk is the event that the kth group works reliably. a) What is the probability that the device will work reliably at P (Zk) = 99%?

There are four independently operating machines in a hall, which do not fail within a certain period of time with the probabilities 0.9, 0.95, 0.8 and 0.85, respectively. Calculate the probability that in this period a) all four machines work b) no machine works c) exactly one machine works d) exactly two machines work e) exactly three machines work f) at least one machine works

Answer 1

There are 100 modules in a device of equal functionality. a) P(Device works reliably)

.

100 P(nk

Since the events $Z_k$'s are independent,

P(n I P(Z) I100.99 (0.99) 100

There are four independently working machines in a hall. Let A, B, C and D be the events that each of them do not fail respectively.

a) as the events are independent

P(AnBn Cn D) = P(A)P(B)P(C)P(D)

0.9 x 0.95 x 0.8 x 0.850.5814

b) P(No machine works) =

P(Ac)P(Be P(C)P(D)= 0.00015

c) P(Exactly one machine works) =

P(A n Ben CnD)P(A n Bencn D) P(A n BnC De)+ P(An Ben cen De)

=0.00565

e) P(exactly three machines work) = 0.34315 (Similar to c)

d) P(exactly two machines work) = 1 - P(exactly four machines work) - P(exactly three machines work) - P(exactly one machine work) - P(no machines work) = 1 - 0.5814 - 0.34315 - 0.00565 - 0.00015 = 0.06965

f) P(At least one machine works) = P(Exactly one machine works) + P(No machines work) = 0.00565 + 0.00015 = 0.0058