A certain lab machine has 22 rings. With each use, these rings can fail, and an

Question

A certain lab machine has 22 rings. With each use, these rings can fail, and an oil leak occurs. The probability of any ring failing during machine use is 8%. The rings are independent, and the failure of one ring does not impact the probability of failing for other rings. The machine gets serviced after each use, so any damaged rings are repaired after each use.

(a) If 4 or more rings fail, the entire machine will shut down. What is the probability of the machine shutting down on any one use? What is the type of probability distribution?

(b) What is the probability that the machine runs successfully at least 5 times before shutting down? What is the type of probability distribution?

Explanation / Answer

Binomial Experiment-
A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
1.The experiment consists of n repeated trials.
2.Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3.The probability of success, denoted by P, is the same on every trial.
4.The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 22 * 0.08
= 1.76
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 22 * 0.08 * 0.92
= 1.6192
III.
standard deviation = sqrt( variance ) = sqrt(1.6192
=1.2725

PART A.
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0) +
= ( 22 3 ) * 0.08^3 * ( 1- 0.08 ) ^19 + ( 22 2 ) * 0.08^2 * ( 1- 0.08 ) ^20 + ( 22 1 ) * 0.08^1 * ( 1- 0.08 ) ^21 + ( 22 0 ) * 0.08^0 * ( 1- 0.08 ) ^22 +   
= 0.90592
P( X > = 4 ) = 1 - P( X < 4) = 0.09408

PART B.
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0) +   
= ( 22 4 ) * 0.08^4 * ( 1- 0.08 ) ^18 + ( 22 3 ) * 0.08^3 * ( 1- 0.08 ) ^19 + ( 22 2 ) * 0.08^2 * ( 1- 0.08 ) ^20 + ( 22 1 ) * 0.08^1 * ( 1- 0.08 ) ^21 + ( 22 0 ) * 0.08^0 * ( 1- 0.08 ) ^22 +
= 0.9727
P( X > = 5 ) = 1 - P( X < 5) = 0.0273

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