I.
Only your missile station can fire 3 missiles at an incoming large
asteroid which will devastate the Earth if it enters the atmosphere intact.
The probability of a hit with the first missile is .4, with the second
it is .5, and with the third it is .7. If the asteroid is hit just
once, then the probability it remains intact is .6; if it it hit twice,
the probability it remains intact is .2, and if it is hit thrice, then
it is certainly blown into small pieces posing no threat to Earth.
What is the probability that Earth will be saved? [.612]
II.
A, B, and C are three sharp shooters. The probability that A hits the
target is 1/3, that B hits it is 1/2, that C hits it is 2/3. Each
of them shoots once at the target. What’s the probability that
there will be exactly one hit? [7/18] If there is one hit only, what’s
the probabaility that it is B’s success? [2/7]
III.
Suppose that the reliability of a chest x-ray to detect tuberculosis
is as follows: of people with tuberculosis, 90% of the x-ray exams detect
the disease, but 10% go undetected. Of people free of tuberculosis,
, 99% of the exams are judged free of disease, but 1% are diagnosed with
tuberculosis. From a large population of which .01% have tuberculosis,
a person is randomly selected and given an chest x-ray. The radiologist
reports the presence of the disease. What’s the probability that
the person has the disease? [.0826]