Please provide suitable solutions to the questions below providing examples where necessary

Question 1:

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

P(T≥t)=e^-t5, for all t≥0.P(T≥t)=e^-t5, for all t≥0.

For example, the probability that the product lasts more than (or equal to) 22 years is P(T≥2)=e^-25=0.6703P(T≥2)=e^-25=0.6703. I purchase the product and use it for two years without any problems. What is the probability that it breaks down in the third year? 

Question 2

Let C1,C2,?,CMC1,C2,?,CM be a partition of the sample space SS, and AA and BB be two events. Suppose we know that

AA and BB are conditionally independent given Ci, for all i∈{1,2,?,M}i∈{1,2,?,M};

BB is independent of all Ci's

Prove that AA and BB are independent.