2.8.4. The law of total probability
Theorem:
Let
B
be an event with
and
.
Then for any event A,
.
Example:
An urn contains 10 white and 6 red balls. Two balls are selected at random without replacement. What is the probability that second selected ball is red?
Solution:
Let
A
be the event that second selected ball is red, B
be event that the first ball is white. Then
,
,
,
.
Then by the law of total probability:
=
.
Theorem:
Let
be
a set of nonempty, mutually exclusive subsets of the sample space S
and
for
then for any event
A of S,
.
Example:
Suppose that 70% of seniors, 60% of juniors, 55% of the sophomores, and 40% of the freshmen of a university use the library frequently. If 35% of all students are freshmen, 30% are sophomores, 20% are juniors, and 15% are seniors, what percent of all students use the library frequently?
Solution:
Let A be the event that a randomly selected student is using library frequently. Let F, O, J, and E be the events that he or she is a freshmen, sophomore, junior, or senior respectively. Thus
.
Therefore, 53% of these students use the library frequently.
Exercises
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 %.