A machine has 9 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.
A)0.086
B)0.066
C)0.980
D)0.914

Question

Quizplus · Accepted Answer

The machine will be working if 0, 1, 2, or 3 components fail. The probability of a component failing is 0.2, and the probability of a component not failing is 0.8. We can use the binomial probability formula, which is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the combination of n items taken k at a time, p is the probability of success, and n is the total number of trials. For 0 failures: P(X=0) = C(9, 0) * (0.2)^0 * (0.8)^9 = 1 * 1 * 0.134217728 = 0.134217728For 1 failure: P(X=1) = C(9, 1) * (0.2)^1 * (0.8)^8 = 9 * 0.2 * 0.16777216 = 0.301989888For 2 failures: P(X=2) = C(9, 2) * (0.2)^2 * (0.8)^7 = 36 * 0.04 * 0.2097152 = 0.30233088For 3 failures: P(X=3) = C(9, 3) * (0.2)^3 * (0.8)^6 = 84 * 0.008 * 0.262144 = 0.176160768Adding these probabilities gives the total probability that the machine will be working: 0.134217728 + 0.301989888 + 0.30233088 + 0.176160768 = 0.914699264, which rounds to 0.914.

A Machine Has 9 Identical Components Which Function Independently