Theorem (Law of Total Probability),
Theorem (Bayesâ€™ Theorem)

**Bayesâ€™
Theorem and its Applications**

In some cases, probability for the occurrence of an event of
interest A may be difficult to compute from the given information. But, it may
be possible to calculate its conditional probabilities *P*(*A*/*B*) and *P*(*A*/) for
some other event *B* of the same
experiment. Then, *P*(*A*) can be calculated applying the law of
total probability. This theorem is a prelude for* *Bayesâ€™ theorem.

In real life situations, decision making is an ongoing process.
Situations may arise where we are interested in an event on an ongoing basis.
Every time some new information may be available and based on this the
probability of the event should be revised. This revision of probability with
additional information is formalized in probability theory in the theorem known
as Bayesâ€™ Theorem.

Let B_{1}, â€¦, B_{n} be n mutually exclusive
events such that where S is the sample space of the random experiment. If P(B_{j})
> 0 for j = 1, 2, â€¦, n, then for any event A of the same experiment with
P(A) > 0,

[This Theorem is due to Rev. Thomas Bayes (1701-1761), an
English philosopher and a priest. This work was published posthumously by his
friend Richard Price during 1763 in the name of Bayes.]

For each event B_{j}, i = 1, 2, â€¦ , n, by the definition
of conditional probability

Mr. Arivazhagan, Mr. Ilavarasan and Mr. Anbarasan attended an
interview conducted for appointing a Physical Teacher in a school. Mr.
Arivazhagan has 45% chance for selection, Mr. Ilavarasan has 28% chance and
Mr.Anbarasan has 27% chance. Also, the chance for implementing monthly Mass
Drill (MD) programme in the school is 42% if Mr.Arivazhagan is appointed; 40%
if Mr.Ilavarasan is appointed; and 48% if Mr.Anbarasan is appointed.

Find the probability that :

i. Mr.Arivazhagan is appointed as the Physical Education
Teacher.

ii. Mr.Ilavarasan is appointed as the Physical Education
Teacher.

iii. Mr.Anbarasan is appointed as the Physical Education
Teacher.

Let MD denote the event that the monthly Mass Drill programme is
implemented in the school. Also, let

A. Mr.Arivazhagan is appointed

B. Mr.Ilavarasan is appointed

C. Mr.Anbarasan is appointed.S

From fig. 8.10 the information given about these events are

P(A) = 0.45

P(B) = 0.28

P(C) = 0.27

P(MD/A) = 0.42

P(MD/B) = 0.40

P(MD/C) = 0.48

With these, the probability for implementing monthly Mass Drill
programme in the school can be computed using total probability as

P(MD) = P(MD/A)P(A)+P(MD/B)P(B)+P(MD/C)P(C)

= (0.42Ã—0.45) + (0.40Ã—0.28) + (0.48Ã—0.27)

= 0.189 + 0.112 + 0.1296

P(MD) = 0.4306.

If it is known that the monthly Mass Drill programme is
implemented in the school,

(i) the probability for Mr.Arivazhagan is appointed as the Physical
Teacher can be calculated applying Theorem8.7 as

(ii) the probability for Mr.Ilavarasan is appointed as the
Physical Teacher can be calculated applying Theorem8.7 as

(iii) the probability for Mr.Anbarasan is appointed as the
Physical Teacher can be calculated applying Theorem8.7 as

Given three identical boxes I, II and III each containing two
coins. In box I, both coins are gold coin, in box II, both are silver coins and
in the box III, there is one gold and one silver coin. A person chooses a box
at random and takes out a coin. If the coin is of gold, what is the probability
that the other coin in the box is also of gold.

In fig 8.11 given below if yellow colour denotes gold coin and
grey colour denotes silver coin then:

Tags : Theorems, Proof, Solved Example Problems , 11th Statistics : Chapter 8 : Elementary Probability Theory

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11th Statistics : Chapter 8 : Elementary Probability Theory : Bayesâ€™ Theorem and its Applications | Theorems, Proof, Solved Example Problems

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