Statistic and Probability discussion

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  1. The probabilities that a parole officer receives 0, 1, 2, 3, 4, or 5 violations a month are 0.15, 0.25, 0.36, 0.18, 0.04, and 0.02, respectively. What are the mean and variance of monthly violations received?

  2. The probabilities that a building inspector will observe 0, 1, 2, 3, 4, or 5 code violations in a newly constructed suburban home are 0.48, 0.25, 0.14, 0.08, 0.04, and 0.01, respectively. What are the mean and variance of building code violations?

  3. Infidelity is given as the reason for 55 percent of all divorce cases filed in Onondaga County. What is the probability that four of the next six divorce cases filed will be due to infidelity?

  4. A study shows that 50 percent of households in Syracuse have at least 2 cars. Out of 16 randomly chosen households, what is the probability that

    1. exactly nine have at least 2 cars;

    2. at most six have at least 2 cars;

    3. anywhere from 8 to 12 have at least 2 cars?

  5. There are ten banks in a country. Each has a 2.0% chance of failing in a given year. The amount of money needed to rescue a failing bank would be $100 million. The government has an insurance fund of $200 million to cover such emergencies. However, if more than two banks fail in a given year, the government cannot pay the depositors, and a general economic panic and collapse will ensue.

    1. Assuming that banks fail independently of each other, what is the chance in any given year that the banking system will collapse?

    2. The assumption that banks fail independently of one another is in fact incorrect. If one bank goes bankrupt, this generally places a strain on the entire banking system, since this bank may default on loans from other banks. So if one bank goes broke, the chance that any one of the others collapses would be greater than 2%. Is the true chance of a catastrophe underestimated or overestimated by the calculation in part a?

  6. A study shows that 40 percent of all patients coming to the emergency room at City Hospital have non-emergency conditions. What are the probabilities that among the next nine patients coming to the emergency room, 0, 1, 2, 3, …, 8, or 9 will have non-emergency conditions?

7. 64 percent of all individuals diagnosed with lymphatic cancer will survive five years beyond the date of diagnosis. Out of 80 newly diagnosed individuals, what is the mean and standard deviation of those who will survive for five years? 

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