# 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.

#### Solution Summary

The solution provides step by step method for the calculation of probability distribution problems . Formula for the calculation and Interpretations of the results are also included.