Medicine and Clinical Studies A patient's blood cholesterol level is often checked during a routine physical examination. A blood sample is taken (from the finger or arm) and tested for total cholesterol and HDL (high-density lipoprotein, the "good cholesterol") cholesterol levels in milligrams per deciliter (mg/dL). If total cholesterol is less than 200 mg/ dL, then no action is taken. If a patient's total cholesterol is approximately 300 mg/dL, a new drug together with a strict diet is prescribed to reduce this number to a safe level. The amount of time (in days) it takes to reduce total cholesterol to a safe level has an exponential distribution with λ = 1/15. A patient with total cholesterol of 300 mg/dL is randomly selected and placed on this drug-and-diet regimen.

a. Find the mean number of days until the total cholesterol level is safe (less than 200 mg/dL).

b. Find the probability that the total cholesterol level will be safe in a number of days within two standard deviations of the mean.

c. Find a value d such that 75% of all such patients have a safe cholesterol level with d days.

d. Suppose two patients are selected at random. Find the probability that this drug-and-diet regimen works for at least one of the patients within 10 days.