Baye's Theorem

BAYE'S THEOREM

[A.U A/M 2019 (R17) RP]

Baye's theorem or Theorem of probability of cases.

Let B₁, B₂, ... Bn be an exhaustive and mutually exclusive random experiments and A be an event related to that B_i then

 

Example 1.3.1

The contents of urns I, II, III are as follows :

One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from urns I, II and III ?   [A.U. M/J 2006, A/M 2008]

Solution:

Let B₁, B₂, B₃ denote the events that the urns I, II, III are chosen respectively and let A be the event that the two balls taken from the selected urn are white and red.

Note :

P(A/B₁) = Probability of getting 1W and 1R balls in urn I

P(A/B₂) = Probability of getting 1W and 1R balls in urn II

P(A/B₃) = Probability of getting 1W and 1R balls in urn III

 

Example 1.3.2

A bag A contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B. [A.U. N/D 2006]

Solution :

Let B₁ the event that the ball is drawn from the bag A

B₂ the event that the ball is drawn from the bag B

A be the event that the drawn ball is red.

 

Example 1.3.3

Companies B₁, B₂ and B3 produce 30%, 45% and 25% of the cars of respectively. It is known that 2%, 3% and 2% of these cars produced for from are defective.

(1) What is the probability that a car purchased is defective ?

(2) If a car purchased is found to be defective, what is the probability that this car is produced by company B₁?

[A.U N/D 2019 (R17) R.P]

Solution:

Let X be the event that the car purchased is defective.

 

Example 1.3.4

A certain firm has plant A, B and C producing IC chips. Plant A produces twice the output from B and B produces twice the output from C. The probability of a non-defective product produced by A, B, C are respectively 0.85, 0.75 and 0.95. A customer receives a defective product. Find the probability that it came from plant B. [A.U. May, 1999]

Solution:

Given: Plant A produces twice the output of B.

Plant B produces twice the output of C.

Let Plant A produces 100 number of IC chips.

Let Plant B produces 50 number of IC chips.

Let Plant C produces 25 number of IC chips.

E → The event that the item produced is non-defective

 → The event that the item produced is defective.

Example 1.3.5

A given lot of IC chips contains 2% defective chips. Each is tested before delivery. The tester itself is not totally reliable. Probability of tester says the chip is good when it is really good is 0.95 and the probability of tester says chip is defective when it is actually defective is 0.94. If a tested device is indicated to be defective, what is the probability that it is actually defective. [A.U. N/D 2004] [A.U N/D 2018 R-17 PS]

Solution: 

 

Example 1.3.6

The members of a consulting firm rent cars from rental agencies. A, B and C as 60%, 30% and 10% respectively. If 9, 20 and 6% of cars from A, B and C agencies need tune up (a) if a rental car delivered to the firm does not need tune up, what is the probability that it came from B agency. (b) if a rental car deliverd to the firm need tune up what is the probability that came from B agency. [A.U. A/M 2004, 2008]

Solution:

Let E₁ be the event that the members of a consulting firm rent cars from rental agency A.

Let E₂ be the event that the members of a consulting firm rent cars from rental agency B.

Let E₃ be the event that the members of a consulting firm rent cars from rental agency C.

 

Example 1.3.7

A binary communication channel carries data as one of 2 types of signals denoted by 0 and 1. Due to noise, a transmitted 0 is sometimes received as a 1 and a transmitted 1 is sometimes received as a 0. For a given channel assume a probability of 0.94 that a transmitted 0 is correctly received as a 0 and a probability of 0.91 that a transmitted 1 is received as a 1. Further assume a probability of 0.45 of transmitted a 0. If a signal is sent, determine the probability that

 (1) a 1 is received

 (2) a 1 was transmitted given that a 1 was received

(3) a 0 was transmitted, given that a 0 was received

(4) an error occurs.

Solution :

 

Example 1.3.8

A box contains 7 red and 13 blue balls. Two balls are selected at random If a and are discarded without their colours being seen. If a third ball is drawn randomly and observed to be red, what is the probability that both of the discarded balls were blue ?  [A.U N/D 2007]

Solution :

Let BB = event that the discarded balls are Blue, Blue

BR = event that the discarded balls are Blue, Red

RR = event that the discarded balls are Red, Red

R = event that the 3rd ball drawn is Red. event

Baye's formula

 

Example 1.3.9

There are 3 boxes containing respectively,

1 white, 2 red, 3 black balls; 2 white, 3 red, 1 black balls;

3 white, 1 red, 2 black balls

A box is chosen at random and from it two balls are drawn at random. The two balls are 1 red and 1 white. What is the probability that they came from second box? [A.U N/D 2019 (R17) PQT] [A.U May 2007]

Solution :

Let B₁, B₂, B₃ denote the events that the boxes are chosen respectively and let A be the event that the two balls taken from the selected box are white and red.

 

Example 1.3.10

A bag contains 3 black and 4 white balls. Two balls are drawn at random one at a time without replacement.

(1) What is the probability that the second ball drawn is white?

(2) What is the conditional probability that the first ball drawn is white if the second ball is known to be white? [A.U A/M 2019 (R17) PQT]

Solution :

Given 3 black balls, 4 white balls

Total number of balls = 3 + 4 = 7

Let A → The first ball drawn is white

B → Second ball is white.

Second ball is white; it can happen in two mutually exclusive ways:

(1) First ball is white and second is white

(2) First ball is black and second is white

 

Example 1.3.11

A consulting firm rents cars from three rental agencies in the following manner 20% from agency D, 20% from agency E one 60% from agency F. If 10% cars from D, 12% of the cars from E and 4% of the cars from F have bad tyres. What is the probability that the firm will get a car with bad tyres? Find the firm will get a car with bad tyres? Find the probability that a car with bad tyres is rented from agency F. [A.U A/M 2019 (R17) PQT]

Solution :

Let A be the event that the car has bad types