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Baye's Theorem

Let us consider 81' 8 2, . .. , Bn are mutually exclusive and exhaustive events such that

B1 + 8 2 + ... + 8n = S.

Let the event A occur in conjunction with only one of the events B1 B2, ... , Bn. If the probabilities P(B1), P(B2), ... , P(B) and P(A/B1), P(A/B2), ... , P(A/Bn) are known, then

2. The probabilities P (Bi) are called 'a priority' probabilities.

 

Example 12. Four boxes Q1, Q2, Q3 and Q4 contain some gold and copper coins. The percentage of the total number of coins in these boxes are respectively 10, 20, 30 and 40. The fractions of gold coins in the boxes are respectively 0.2, 0.3, 0.1 and 0.5.

(i) If a coin is taken out at random, what is the probability that it is a gold coin?

(ii) If a coin is taken out at random is found to be golden, what is the probability that it is ·taken from box Q2?

            Solution. Let   P(G) = Probability that the coin is a gold coin

                                    P(Q) = Probability that the coin come from box Or

 

Example 13. In a factory manufacturing bulbs, machines numbered 1, 2, 3 manufacture respectively 20, 45 and 35 percent of the total output. Of their outputs 3, 5 and 4 percent respectively are defective. A bulb is drawn at random from the total output and is found to be defective. Find the probability that it was manufactured by the machine numbered.

 

Solution. Let Mi = Event that the bulb is produced by machine i. (i = 1 ,2,3)

           P(M1) = 0.2,     P(M2) = 0.45, P(M3) = 0.35

 

Let                   A = Event that the bulb is defective.

                        P (A/M1) = 0.03, P(A/M2) = 0.05, P(NM3) = 0.04.

 

Then using Baye's theorem we calculate the following probabilities.

P(machine 1 producing defective bulb)

 

P (M2/A) = 0.53 and P (M3/ A) = 0.33.

 

Example 14. Two persons A and B fire at a target independently and have a probability 65 and 0. 72 respectively of hitting the target. Find the probability that the target is destroyed.

 

Solution.                     Let E1 = Event that target is hit by A

E2 = Event that target is hit by B.

Given                           P(E1) = 0.65, P(E2) = 0.72

P(E1) = Probability of failure to hit by A

= 1-P(E1)=0.35

and                                         P(E2 ) = 1 - P(E2) = 0.28

 

P(Both of them fails)          =          P(E1). P(E2 )                    =                  0.35 x 0.28 0.098

Therefore, P(Target is destroyed)

 

Probability that at least one o~ them hit

1 - P(Both of them fails)

Alternative Method.

P(Target is destroyed)P(E1+E2)

=           P(E1) + P(E2) - P(E1E2)

=             P(E 1) + P(E2) - P(E 1 ) · P(E2)

=                0.65 + 0.72 - (0.65 X 0.72)         =              0.902.

 

PROBLEMS

(i) A and B are mutually exclusive, (ii) A and B are independent.

 

4. Out of the numbers 1 to 100, one is selected at random. What is the probability that it is divisible by 7 or 8?

 

5. A committee of 4 people is to be appointed from 3 offices of the production department, 4 officers of the purchase department, 2 officers of the sales department and I chartered accountant. Find the probability of forming the committee in the following manner.

(i) There must be one from each category.

(ii) The committee should have at least one from the purchase department.

(iii) The chartered accountant must be in the committee.

 

6. What is the probability that a leap year selected at random will contain either 53 Thursdays or 53 Fridays?

 

7. A candidate is selected for interview for three posts. For the first post there are 6 _candidates, for the second there are 9 candidates and for the third there are 5 candidates. What are the chances for his getting at least one post?

 

8. The odds that person A speaks the truth are 3: 2 and the odds that person B speaks the truth are 5: 3. In what percentage of cases are they likely to contradict each other in stating the same fact?

 

9. In an examination, 30% of the students have failed in Engineering Mechanics, 20% of the students have failed in Mathematics and I 0% have failed in both the subjects. A student is selected at random.

 

(i) What is the probability that the student has failed in Engineering Mechanics if it is known that he has failed in Mathematics?

(ii) What is the probability that the student has failed either in Engineering mechanics or in Mathematics?

 

10. X can solve 80% of the problems while Y can solve 90% of the problems given in a Statistic book. A problem IS selected at random. What is the probability that at least one of them will solve the same?

 

11. A box P has 1000 items of which 100 are defective. Another box Q has 500 items of which 20 are defective. The items of both the boxes are mixed and one item is randomly taken out. It is found to be defective. What is the probability that the item belongs to box P?

 

12. Villages A, B, C and D are connected by overhead telephone lines joining AB, AC, BC, BD and CD. As a result of severe gales, there is a probability p (the same for each link) that any particular link is broken. Find the probability of making a telephone call from A to B.

 

13. A man seeks advice regarding one of two possible courses of action from three advisers, who arrive at their recommendations independently. He follows the recommendation of the majority. The probabilities that the individual advisers are wrong are 0.1, 0.05 and 0.05 respectively. What is the probability that the man takes incorrect advice?

 

14. An unbiased coin is tossed four times in succession and a man scores 2 or 1 according to the coin as shows head or tail in each throw. El' E2 are the following events: E 1 : total score is even, E2 : total score is divisible by 3. Determine whether E1 and E2 are independent or not.

 

15. There are two identical boxes containing ' respectively 4 white and 3 red balls and 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is white

If the ball is white, what is the probability that it is from the first box?

 

16. A speaks truth 4 out of 5 times. A die is tossed. He reports that there is a six. What is the chance that actually there were six?

 

17. If a machine is set correctly it produces 10% defective items. If it is set incorrectly then it produces 10% good items. Chances for a setting to be correct and incorrect are in the ratio 7: 3. After a setting is made, the first two items produced are found to be good items. What is the chance that the setting was correct?

 

18. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of accident is 0.01, 0.03 and 0.15 respectively. One of the insured persons meets an accident, what is the probability that he is a car driver?

 

19. A bag has 5 red and 4 green balls; a second bag has 4 red and 6 green balls. One ball is drawn from the first and two from the second. Find the probability that out of three balls

(i) All three balls are red,

(ii) All three balls are green,

(iii) Two are red and one is green.

 

20. Suppose that if a person travels to India, the probability that he will see Delhi is 0.70, the probability that he will see Kolkata is 0.64, the probability that he will see Hyderabad is 0.58, the probabilities that he will see Delhi and Kolkata is 0.42, Delhi and Hyderabad is 0.51, Kolkata and Hyderabad is 0.40 and the probability that he will see all the three cities is 0.21. What is the probability that a person travelling to India will see at least one of these three cities?

 

21. Urn I contains 2 white and 4 black balls and urn II contains 4 white and 4 black balls. If a ball is drawn at random from one of the two urns, what is the probability that it is a white ball?

 

22. A study of daily rainfall at a place has shown that in July, the probability of a rainy day following a rainy day is 0.4, a dry day following a dry day is 0.7, a rainy day following a dry day is 0.3 and a dry day following a rainy day is 0.6. If it is observed that a certain July day is rainy, what is the probability that the next two days will also be rainy 'I

 

23. The inflow in a cylindrical water tank in a house is equally likely, to fill 6, 7 or 8 ft. of the tank. According to the demand the outflow is 5, 6 or 7 ft. suppose the water level at the tank is 6 ft. at the start of the day.

(i) What is the possible water level in the tank at the end of the day?

(ii) What is the probability that there will be at least 8 ft. of water remaining in the tank at the end of the day?

 

ANSWERS

 

1. p(B)=2/7                              2.p(A/B)=1/2, p(AUB0=12    Independent

 

3. (i)Not mutually exclusive,    (ii) independent                                   4. 1/4

 

5.(i) 8/70                      (ii) 13/14                      (iii) 2/5             6.3/7

 

7. 11/27                                                           8. 47.5%of cases

 

9.(i) 1/2                        (ii) 0.40                        10. 49/50         11. 5/6

 

12.1-2P2+P3                 13. 0.01175                 14. Not Independent

 

15. 61/140,40/61         (use bayes theorem)

 

16. 4/9(Use Baye's theorem)               17. 0.99(Use Baye's theorem)

 

18. 0.12 (Use Bayes' theorem)            19.(i). 2/27 , (ii)4/27    (iii).16/45

 

20. 0.80                                               21.0.4166                    22. 0.16

 

23. (i) 5, 6, 7' 8, 9 ft., (ii)- 113.