Question

The lifetime of a device behaves according to the probability law for P(T>t) = 1/t for t>1. Let A be the event 'lifetime is greater than 4,' and B the event 'lifetime is greater than 8.' Using the provided probability law: a) Determine the probability of event A, P(A).

b) Determine the probability of event B, P(B).
c) Determine the probability of the intersection of events A and B, P(A∩B).
d) Assess whether events A and B are independent, dependent, or mutually exclusive. Provide a justification for your conclusion.

Answer 1

The probability of the events A 'lifetime greater than 4' and B 'lifetime greater than 8' were calculated based on a given probability law and analyzed to determine if they are independent, dependent, or mutually exclusive.

Step-by-step explanation:

The student's question is regarding probability and how to calculate probabilities of specific events when given a probability law. The events pertain to the lifetime of a certain device.

1. P(A), the probability of the event 'lifetime greater than 4', is found by substituting 4 into the given probability law.
2. P(B), the probability of the event 'lifetime greater than 8', is also found through substitution into the law.
3. P(A∩B), is equivalent to P(B), as B is wholly contained within A. The intersection of A 'lifetime greater than 4' and B 'lifetime greater than 8' is effectively just 'lifetime greater than 8'.
4. For independence, if P(A ∩ B) = P(A)P(B), the events A and B are independent. We compare the calculated probabilities to assess if this condition is met.
5. For mutual exclusivity, no elements are shared between events; however, for dependency, elements of event A influence the probability of event B.