Suppose that a foreman must select one worker from a pool of four available work

Question

Suppose that a foreman must select one worker from a pool of four available workers (numbered I,2,3, and 4) for a particular job. He selects the worker by mixing the four names in a hat and randomly drawing one (i.e.. each person is equally likely to be selected). Define the following events: A = the event that either worker 1 or worker 2 is selected B = the event that either worker 1 or worker 3 is selected C = the event that worker 1 is selected D) = the event that worker 3 or worker 4 is selected Answer the following questions and explain why or why not (a) Are events A and B independent? (b) Are events A and C independent? (c) Are events A and D independent?

Explanation / Answer

Since each worker is equally likely to select so the probabaility of selecting any worker is 1/4. Since either worker 1 or worker 2 can be selected but not both so the probability of event A is

P(A) = (1/4) + (1/4) = 1/2

Likewise

P(B) = (1/4) + (1/4) = 1/2

P(C) = 1/4

P(D) = (1/4) + (1/4) = 1/2

(a)

Event A and event B both can be happened of worker 1 is selected so

P(A and B) =1/4

Since

P(A and B) = P(A)P(B)

So A and B are independent.

(b)

Event A and event C both can be happened of worker 1 is selected so

P(A and C) =1/4

Since

P(A and C) = P(A)P(C)

So A and C are independent.

(c)

Since event A and D cannot be occur simultaneously so events A and D are mutually exclusive. So

P(A and D) = 0

That is event A and D are not independent.

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