In genetic profiling, the expression of a gene is measured for a set of different samples. Suppose that the expressions are modeled as being independently normally distributed with a mean 0.768 and a standard deviation 0.083.
(a) If six samples are measured, what is the probability that at least half of them have expressions larger than 0.800?
(b) If six samples are measured, what is the probability that two of them have expressions smaller than 0.700, two of them have expressions between 0.700 and 0.750, and the remaining two have expressions larger than 0.780?
(c) If the samples are tested sequentially, what is the probability that the sixth sample tested is the third sample with an expression smaller than 0.760?
(d) If the samples are tested sequentially, what is the probability that the fifth sample tested is the first sample with an expression smaller than 0.680?
(e) Suppose that 10 samples are tested and exactly 5 of them have expressions smaller than 0.750. Furthermore, 6 samples are randomly selected from these 10 samples and are sent to another laboratory. What is the probability that exactly half of the samples sent to the other laboratory have an expression smaller than 0.750?