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QUESTION

Staying Employed-Conditional Probability with explanation

Due to budget cuts a manager was told to cut a single person out of his work crew of 4 workers-Dave, Joe, Mike, and Roger.  He decided to make the cut in as fair of way as possible since all are equally qualified.  He put each of their names on a card, mixed the cards, and drew a random name.  The manager’s best friend is Dave and Dave stated that at least his odds of losing his job are only one out of four-“it could be worse”, Dave stated.  Then Dave asked, “At least two of the others-Joe, Mike, and Roger will retain their job; could you tell me the names of the two that will keep their jobs?”  The manager decided to tell Dave which of the others would keep their jobs, the manager stated, “Mike and Roger will keep their jobs”…..Dave’s concern set in, he said it seems as if my chances of being terminated just went from ¼ to ½……..

Is John right in thinking that the probability changed from ¼ to ½?  Explain/prove your answer with a decision tree or via formulas....

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