MADE EASY TEST SERIES

Question

You purchase a certain product. The manual states that the life­time T of the
product, defined as the amount of time (in years) the product works properly
until it breaks down, satisfies

P(T >= t) = e^{(-t/5) ,for all t>=0}
you purchase the product and use it for two years without any problems. The
probability that it breaks down in the third year is _________ (upto three
decimal places).
plz solve it.

Comments

What is the answer given by the way ??

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ans is .181

Answer 1

Basically the probability being asked is conditional probability :

P((2 <= x <= 3) | (x >= 2) ) meaning that it will break somewhere in the third year but we are given that the product is well till 2 years .The numerator term of probability can be written as :

P(2 <= x <= 3) ∩ (x >=2 )) =  P(2 <= x <= 3 ) 

                                       =  P(x >= 2)  - P(x >= 3)

                                       =  1/e^2/5 -   1/e^3/5  (according to the given function for P(x >= t) = e^-t/5)

                                       =  0.1215

But we are given it works fine till 2 years .

So  ,   P(x >= 2 ) = 1 / e^0.4 

                          = 0.6703

Therefore , P((2 <= x <= 3) | (x >= 2) )   =    P(2 <= x <= 3) ∩ (x >=2 )) / P(x >= 2 )

                                                          =    0.1215 / 0.6703

                                                          =    0.181

Hence the required probability should be 0.181

Comments

thanks ..i got it.

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In this question we can't take it as a random variable and think it as a cdf and find P(2<=x<=3) =F(3)-F(2) and den minus it from 1 .?? Why can't we do like this sir