A mcg of U-238 emits particles according to a Poisson process at rate 2/sec. A mcg of U-235 emits particles according to a Poisson process at rate 10/sec. Marie receives a gift of 1 mcg uranium, which is U-238 with probability 0.5 and U-235 with probability 0.5.
(a) Marie observes the gift for one second. What is the probability that no particles are emitted?
(b) Suppose Marie observes that 15 particles are emitted from her mcg of uranium during a 3 second period. Conditioned on that observation, what is the probability that the uranium is U-235?
(c) For this question, do not suppose Marie observed 15 particles emitted in 3 seconds on the first day. The next day, Marie receives a second gift of 1 mcg uranium, of the type she didn’t receive previously. She observes both gifts simultaneously. The first particle emitted by either onecomes from the first. What is the probability that the first is U-235? If the first two particlesemitted come from the first, what is the probability that the first is U-235?
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