In 1986, one of the largest nuclear power plants in the Ukraine suffered a horrible explosion that sent radioactive iodine and cesium into the air. A large amount of radiation escaped into the atmosphere, and much is still contained in the soil. The half-life of cesium-137 is about 30.2 years. If a soil sample was found to contain 18.4 mg of cesium-137 in 2005, how much was released into the soil at the time of the accident? How much is cesium-137 will be left in 2030?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.