Exercises. 1. In a country men constitute 58% of the labour force

1. In a country men constitute 58% of the labour force. The rates of unemployment are 6.2% and 4.3% among males and females respectively.

a) What is the overall rate of unemployed in the country?

b) If a worker selected at random is found to be unemployment, what is the probability that the worker is a woman?

2. In a shipment of 15 air conditioners, there are 4 with defective thermostats. Two air conditioners will be selected at random and inspected one after another. Find the probability that

a) The first is defective.

b) The first is defective and the second good.

c) Both are defective.

d) The second air conditioner is defective.

e) Exactly one is defective.

3. Suppose that 40% of the students are girls. If 25 % of the girls and 15% of the boys of this university are A students, what is the probability that randomly selected student is A student?

4. A factory produces all its products by three machines. Machines I; II;

and III produces 40%; 40% and 20% of the output, where 5%, 4%, and 2% of their outputs are defective, respectively. What percentage of the total product is defective?

5. A box contains 18 tennis balls, of which eight are new. Suppose that three balls are selected randomly, played with, and after play are returned to the box. If another three balls are selected for a second play, what is the probability that they are all new?

6. In an economical college all students are required to take calculus and economics course. Statistics shows that 37 % of the students of this college get A’s in calculus and 25 % of them get A’s in both economics and calculus. If randomly selected student of this college has passed calculus with an A, what is the probability that he or she got A in economics?

7. Suppose that 12 % of the population of a country are unemployed women and 17 % of population are unemploymed. What percentage of the unemployed are women?

Answer

1. a) 5.4%; b) 0.334; 2. a) 4/15; b) 22/105; c) 2/35; d) 4/15; e) 44/105;

3. 0.19; 4. 4%; 5. 0.148; 6. 0.676; 7. 70.6 %.

2.9. Bayes’ theorem

Often, we begin our analysis with initial or prior probability estimates for specific events or interest. Then, form sources such as a sample, a special report, a product test, etc., we obtain some additional information about the events. Given this new information, we want to revise and update the prior probability values. The new or revised probabilities for the events are referred to as posterior probabilities. Bayes’ theorem, which will be presented shortly, provides a means of computing these revised probabilities. To introduce Bayes’ formula, let us consider the following example:

Example 1:

In the factory 40%, 30%, and 30% of the goods is produced by machines I, II, and III, respectively. If 5%, 4%, and 3% of the outputs of these machines is defective, what is the probability that a randomly selected good that is found to be defective is produced by machine III?

Solution:

Let A be the event that a randomly selected good is defective and be the event that it is produced by machine III. We are asked to find . We know that

To find , note that since and are known, we can use relation .

To calculate , we use the law of total probability. Let and be the events that the good is produced by machines I and II, respectively. Hence,

By substituting we obtain

.

The formula for is a particular case of Bayes’ formula. To write formula for we can use tree diagram. (Fig. 2.1). Letter D stands for “defective” and N for “not defective”.

Theorem: (Bayes’ theorem)

Let A and B be two events. Then Bayes’ theorem states that

and

Theorem (Bayes’theorem, general form):

Let be n mutually exclusive and collectively exhaustive events of the sample space S. Then for any other event A of S with

Example 2:

A box contains 8 red and 11 blue balls. Two balls are selected at random without replacement and without their colour being seen. If the third ball is drawn randomly and observed to be red, what is the probability that both of previous selected balls were blue?

Solution: Let BB, BR, and RR be the events that first two selected balls are blue and blue, blue and red, and red and red. Let R be the event that the third ball drawn is red. We need to find . Using Bayes’ formula:

.

Now

and

,

where BR is the union of two events:

namely, the first ball was blue, the second was red, and vice versa.

Thus,

.

This can be found easily from tree diagram on Fig. 2.2. as well.

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