Random variables and probability problems

1. Seventy percent of the light aircraft that disappear while in flight in a certain
country are subsequently discovered. Of the aircraft that are discovered,
60% have an emergency locator, whereas 90% of the aircraft that are not
discovered do not have an emergency locator. Suppose a light aircraft has
disappeared.
a) Draw a corresponding tree.
b) Find the probability that it will be discovered?
c) If it has an emergency locator, what is the (conditional) probability
that it will be discovered?

2. Seventy percent of all vehicles examined at a certain emissions inspection
station pass the inspection. Assume that successive vehicles pass or fail
independently of one another.
(a) Calculate P(all of the next three vehicles inspected pass)
(b) Calculate P(at least one of the next three inspected fail)
(c) Calculate P(exactly one of the next three inspected passes)
(d) 10 cars will be tested (one after another) in the following day. Find the
probability that at least five cars will be tested to have a car which does not pass
the inspection.

3. An inspector is testing (one after another) batteries of certain type until he
finds a defective one. The proportion of defectives is 0.05 and we assume
that the tests are independent.
a) Find probability then more than 4 tests will be required. Hint: What
is the complement to the above event?
b) Let Y = # of tests required to find the first defective. What are
possible values of Y.
c) Find the pmf of Y.
d) Now suppose that he tests until he finds the second defective. Find
probability that exactly 5 tests will be required.

4. There are four batteries of a particular type in a package. The distribution of a random
variable X - the number of defective batteries in each package is

X 0 1 2 3 4
p(x) 0.8 0.1 0.05 0.03 ?

a) Find the missing value.
b) What is the probability that there are at most 2 defective batteries in a
packages?
c) Find the probability that there is at least one defective in a package.
d) Find and graph cdf of X.