Baye's Theorem

BAYE'S THEOREM

Baye's theorem or Theorem of probability of cases. [A.U A/M 2019 (R17) RP]

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

Proof :

According to conditional probability

Using multiplication rule of probability

Using total probability theorem

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.

Then

 

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.

P(B₁) = P(B₂) = 1/2

Example 1.3.3

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

Ē -> The event that the item produced is defective.

Hence P(A) = 100/175 = 0.57; P(B) = 50/175 = 0.29; P(C) = 25/175 = 0.14

Given : The probability of a non-defective product produced by A, B and C are respectively 0.85, 0.75 and 0.95

To find the probability that the customer receives a defective product from plant B.

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 :

E -> Event of chip is actually good.

Ē-> Event of chip is actually defective

we know that P (E) + Ē = 1

D -> Event of tester says it is good.

 -> Event of tester says it is defective.

Given: Lot of IC chips containts 2% defective chips.

Given: Probability of tester says the chip is good when it is really good is 0.95.

Given: Probability of tester says the chip is defective when it is actually defective is 0.94.

To find The probability of actually defective

i.e., To find 

By Baye's theorem

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.

Given: P (E1) = 60% = 60/100 = 0.60

Let D be the event that cars need tune up.

Let  be the event that cars need not tune up.

(a) To find 

(b) To find 

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. ([A.U M/J 2007]

Solution:

A event of transmitting a zero

B - event of receiving a zero P (A) = 0.45

(1) P (1 is received)

Example 1.3.8

A box contains 7 red and 13 blue balls. Two balls are selected at random 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.

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.

Then

Baye's theorem

Hence, 

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) PCT]

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

(1) 

(2) To find the conditional probability P (A/B)

P (A/B) = P (A∩B)/P(B) .............(2)

A∩B = first ball was white and second ball is also 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 Satidw ad of agency F. [A.U A/M 2019 (R17) PQT]

Solution :

Let A be the event that the car has bad types

Given:

P(A) = P(D) P(A/D) + P(E) P(A/E) + P(F) P(A/F)

(b) P(F/A) = P(A∩F)/P(A) = P(F) P(A/F)/P(A)

EXERCISE 1.2 and EXERCISE 1.3

1. Prove that if P(A) > P(B) then P(A/B) > P(B/A)

2. From 5 red balls and 8 white balls, 4 balls are chosen at random (without replacement) find the probability that they are of the red colour. [Ans. 1/143]

3. A card is drawn from a well shuffled pack of 52 cards. What is the probability that it is either clever or king. [Ans. 4/13]

4. A box contains 4 bad and 6 good tubes. Two are drawn out together. One of them is tested and found to be good. What is the probability that the other one is also good. [Ans. 5/9]

5. It is 8 5 against a person who is now 40 years old living till he is 70 and 4.3 against a person now 50 living till he is 70. Find the probability that atleast one of these persons will be alive 30 years. 

6. Given P(A) = 0.50, P(B) = 0.30 and P(A∩B) = 0.15 Verify that

7. A certain item is manufactured by three factories say 1, 2 and 3. It is known that 1 turn out twice as many item as 2 and that 2 and 3 turns out the same number of item (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective, while 4 percent of those manufactured by 3 are defective. All the items produced are but into one stockpile, and then one item is chosen at random. What is the probability that this item is defective? [Ans. 0.025]

8. A box contains five balls. The balls are drawn and found to be white. What is the probability that all the balls being white. [Ans. 1/4]

9. Two students A and B work independently on a problem. The probability that the first one will solve it is 3/4 and the probability that the second one will solve it is 2/3. What is the probability that the problem will be solved? [Ans. 11/12]

10. An urn contains 5 white and 3 green balls and another urn contains 3 white and 7 green balls. Two balls are chosen at random from To the first urn and put into the second urn, Then a ball is drawn from the second urn, what is the probability that it is a white ball? [Ans. 0.372]

11. Ten chips numbered 1 to 10 are mixed in a bowl. Two chips are drawn from the bowl successively and without replacement. What is the probability that their sum is 10? [Ans. 4/45]

12. A bag contains 10 tickets numbered 1, 2, ... 10. Three tickets are drawn at random are arranged in ascending order of magnitude. What is the probability, that the middle number is 5 ? [Ans. 1/6]

13. Four persons are chosen at random from a group consisting of 4 men, 3 women, and 2 children. Find the chance that the selected group contains at least one child. [Ans. 13/18]

14. A box contains 3 white balls and 2 black balls. We remove at random 2 balls in succession, what is the probability that the first removed ball is white and the second is red ?

6/25 (if the ball is replaced)

3/10 (if the ball is not replaced)

15. An urn contains ten white and three black balls. Another urn contains three white and five black balls. Two balls are drawn at random from the first urn and placed in the second urn and there one ball is taken at random from the latter. What is the probability that it is a white ball? [Ans. 59/130]

16. Urn I has two white and three black balls. Urn II has four white island one black balls and urn III has three white and four black balls. An urn is selected at random and a ball drawn at random is found to be white. Find the probability that urn I was selected. [Ans. 14/57]

17. A bag contains seven red and three black marbles and another bag contains four red and five black marbles. One marble is [ST transferred from the first bag into the second bag and then a marble is taken out of the second bag at random. If this marble happens to be red, find the probability that a black marble was transferred. [Ans. 12/47]

18. There are three true coins and one false coin with head on both sides. A coin is chosen at random and tossed four times. If head occurs all the four times, what is the probability that the false coin has been chosen and used ? [Ans. 16/19]

19. Three urn contains three white, one red and one black balls; two 181 white, three red and four black balls; one white, three red and two black balls respectively. One urn is chosen at random and 16 from it two balls are drawn at random. If they are found to be 12 one red and one black balls, what is the probability that the first urn was chosen? [Ans. 3/25]

20. A bolt is manufactured by three machines A, B, C. A turns out twice as many items as B and machines B and C produce equal number of items. Two percent of bolts produced by A and B are defective and four percent of bolts produced by C are defective. All bolts are put into one stock pile and one is chosen from this pile. What is the probability that it is defective. [Ans. 1/40]