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# Probability distribution that you will receive a sample mean grand prize

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Game show shows you a large revolving drum containing four identical white envelopes that have been thoroughly mixed in the drum. Each of the envelopes contains one of four decks made out for grand prizes of 20, 40, 60, and 80 thousand dollars. Usually, a contestant reaches into the drum, selects and envelope, and receives the grand prize in the envelope. Tonight, however, is a special night. You will be given the choice of selecting one envelope or selecting two envelopes and receiving the average of the grand prizes in the two envelopes. If you select one envelope, the probability is ¼ that you will receive any one of the grand prizes 20, 40, 60, and 80 thousand dollars.

Find the probability distribution of the population that you will receive a sample mean grand prize of at least 50 thousand dollars?