Question

Suppose 4 components of system all act independently, and each fails within a year 10% of the time. Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.

Answer 1

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SOL ;;-

AS FOR GIVEN DATA....

number of components are n=4, probability that a component will fail is p=0.1

X: Number of failures. Then X~ Binomial(n=4, p=0.1)

The Probability Mass Function of Binomial distribution is

PlX-z)-( :) pr . (1-p)"-' ; に0,1, 2, 3, 4,

(1 ) The probability that none fail within a year is' ?

SOL:

  

. C) P(X - 0) 0.1" * (1-0.1)4-0 0.94-0.6551

(2) The probability that at least 1 fails within a year is ?

SOL::

| ) 0.11 × 0.94-1-( ) 0.12 × 0.94-2 ) 0.13 × 0.94-3 :Γ ) 0.14 ×0.94-1 = 0.2916 0.0486 + 0.0036 + 0.0001 = 0.3439

(3) The probability that all fails within a year is ?

SOL::

Pr(X = 4) = (: ) 0.14 x 0.94 | ) 0.14 . O.94-4-0.14 = 0.0001 Pr(X 4)

(4)   Here we use p=1-0.1=0.9 and X: Number of working components.

The probability that at least 1 works entire year is ?

SOL ::

= 0.0036 + 0.0486 + 0.2916 + 0.6561 0.9999

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