Answer :

Given that, a certain kind of component will survive a given shock


Let p be the probability that component survives the shock test.


Then,



If p is the probability that component survives the shock test, then q is the probability that the component doesn’t survive the shock test.


p + q = 1


q = 1 – p





Let X denote a random variable that represents the components that survive the shock test out of the 5 components tested.


The probability that r components out of n components survive the shock test is given by the binomial distribution.


P (X = r) = nCrprqn-r


Here, n = 5 (sample components tested)



And


We can re-write it as,


…(A)


(i). We need to find the probability that among 5 components tested exactly 2 will survive.


Then, put r = 2 in equation (A). We have









P (X = 2) = 0.0879


, the probability that exactly 2 components out of 5 will survive the test is 0.0879.


(ii). We need to find the probability that among the 5 components tested at most 3 will survive.


So, this can be give in two ways:


(a). Probability that out of 5 components at most 3 will survive = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)


(b). Probability that out of 5 components at most 3 will survive = 1 – [P (X = 4) + P (X = 5)]


Let us solve it by using the formula in (b).


So, let us find out P (X = 4).


Putting r = 4 in equation (A), we get






Now, let us find out P (X = 5).


Putting r = 5 in equation (A), we get





Using the values of P (X = 4) & P (X = 5) in formula (b), we get









Probability = 0.3672


, the probability that out of 5 components at most 3 will survive is 0.3672.


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