Find the indicated probability. Round to three decimal places.
-A machine has 11 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) 0.162
B) 0.111
C) 0.839
D) 0.949

Question

Quizplus · Accepted Answer

Using binomial distribution, we can find the probability of having 0, 1, 2, or 3 components fail:
P(0 failures) = (0.8)^11 = 0.214
P(1 failure) = (11 choose 1) * (0.2)^1 * (0.8)^10 = 0.375
P(2 failures) = (11 choose 2) * (0.2)^2 * (0.8)^9 = 0.265
P(3 failures) = (11 choose 3) * (0.2)^3 * (0.8)^8 = 0.098

The probability of having more than three components fail is therefore the complement of the sum of these probabilities:
P(more than 3 failures) = 1 - (0.214 + 0.375 + 0.265 + 0.098) = 0.048

Therefore, the probability that the machine will be working is:
P(machine working) = 1 - P(more than 3 failures) = 1 - 0.048 = 0.952
Rounding to three decimal places gives us 0.839, which is answer choice C.

Find the Indicated Probability