Probability Exercises - Weebly



Probability Exercises

1. Suppose we have a bag with 3 red,2 black and 5 white balls. From this bag a ball is drawn and it is returned . What is the probability of drawing

( i ) either a red or a black ball ?

( ii ) either a white or a black ball ?

2. A sub committee of 6 members is to be formed out of a group consisting of 7 men and 4 women. Calculate the probability that the sub committee will consist of ( a ) exactly 2 women and ( b ) at least 2 women ?

3. The probability that A can solve a problem is 4/5, B can solve it is 2/3, and C can solve it is 3/7.If all of them try independently find the probability that ( a ) Problem will be solved ( b ) Problem will not be solved.

4. A bag contains 6 red and 5 blue balls and another bag contains 6 red and 8 blue balls. A ball is drawn from the first bag and without noticing its colour put into the second bag. A ball is then drawn from the second bag. Find the probability that ball drawn is blue.

5.Two cards are drawn from a pack of 52 at random and kept out. Then one card is drawn from the balance 50 cards. Find the probability that it is an ace.

6.The odds against X solving a problem are 8 to 6.The odds in favour of Y solving are 14 to 16. What is the probability that the problem( a ) will not be solved ( b ) will be solved.

7. A coin is tossed 6 times . What is the probability of getting at least 2 heads.

8. Two persons throw a dice alternately till one of them gets a multiple of 3 and wins the game. Find there respective chances of winning.

9. A man simultaneously tosses a coin and throws a die beginning with the coin. What is the probability that he will get a head before he gets a 5 or 6 .

10. Three machines A,B,C produce respectively 50%,30%,20% of the total no of items in a factory. The percentages defective items by these machines are 3%,4%,5% respectively. An item is drawn at random find the probability that item is defective .

11.The probabilities of X,Y,Z becoming managers are 4/9,2/9,1/3 respectively. The probabilities that bonus scheme will be introduced if X,Y,Z, becomes manager are 3/10,1/2,1/5 respectively.

( a ) What is the probability that bonus scheme will be introduced? ( b ) If the bonus is introduced the manager is X.

12. There are 4 boys and 2 girls in room A, and 5 boys and 3 girls in room B. A girl from one of the two room laughed loudly. What is the probability that girl is from room B

13. You observe that your boss is happy with your call is 0.6. When he is happy the probability of acceding to your request is 0.4 whereas if he is unhappy probability of acceding to your request is 0.1.You call one day and he accedes to your request . What is the probability of boss being happy?

14. In a competition an examinee either guesses or copies or knows the answer to a multiple choice question with 4 choices. The probability that he makes a guess is 1/3. he copies is 1/6, and he knows is 1/8. Find the probability that he knew the answer to the question ,given that answer is correct.

15. A company uses selling aptitude test in the selection of salesman. Past experience has shown that only 70% of all persons applying for a sales position achieved a classification of dissatisfactory in actual selling , whereas the remainder were classified satisfactory. 80% has scored a passing grade on the aptitude test. Only 25% of those classified dissatisfactory has passed the test on the basis of this information . what is the probability that a person would be satisfactory salesman given that he passed the aptitude test .

16.A candidate is selected for interview for 3 posts . For the first post there are 3 candidates, for the second post there are 4 and for the third there are 2 candidates. What are the chances that the candidate gets at least one post.

17. The table below gives the probabilities of error and number of account statement made by A,B,C respectively.

A B C

Probability 0.2 0.25 0.55

Number 10 16 20

Find the expected number of correct statements in all.

18. A tennis player plays two sets against an opponent. He has 50% chance of winning first set, and, if he wins the first, he has a 55% chance of winning the second set, But if he loses the first, he has only 40% chance of winning the second set.

( i )What is the probability of winning both sets?

( ii ) If you are told he lost the second set , what is the probability he lost both sets ?

19.A dice is tossed twice . Getting a number greater than 4 is considered success. Find the mean and variance of probability of number of successes.

20. Two bad oranges are accidentally mixed with ten good oranges . Three oranges are drawn at random without replacement from this lot. Find the mean and variance of number of bad oranges.

21.A purchase Manager has placed order for a particular Raw material with two suppliers, A and B . The probability that supplier A can deliver in 4 days is 0.55 . The probability that supplier B can deliver in 4 days is 0.35 .

i) What is the probability that both suppliers will deliver the material in 4 days ?

ii) What is the probability that at east one supplier will deliver the material in 4 days ?

iii) What is the probability that none will deliver the material in 4 days ?

22 A bank is reviewing its credit card policy for canceling some cards. In the past approximately 5% of card holders have defaulted and bank has been unable to collect the outstanding balance. Hence management has attached prior probability of 0.05 that any particular cardholder will default. The bank has further found that the probability of missing one or more monthly payments is 0.20 for customers who do not default. The probability of missing one or more monthly payments for those who default is 1.

i) Given that a customer has missed a monthly payment , compute the posterior probability that the customer will default.

ii) The bank would like to cancel the card if the probability that a customer will default is greater than 0.20.Should the bank cancel the card if the customer misses a monthly payment ? why or why not ?

23. Consider the possibility of a dangerous radioactive leakage occurring at a nuclear power generating plant. Suppose that a device designed to warn against leakage detects a problem in sufficient time to take corrective action with probability 0.98. Also, the probability that the device indicates action should be taken when no problem exists is 0.10. An environmentalist claims that the chance of leakage is 0.03 at any point of time. What is the probability that a leakage problem actually exists if the warning device now indicate this ?

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| |prior prob |condl prob |jt prob |posterior prob |

| |P(event) |P(outcome/event) |P ( outcome&event) |P(event/outcome) |

|L |0.03 |0.98 |0.0294 |0.232594937 |

|L Bar |0.97 |0.1 |0.097 |0.767405063 |

| | | |0.1264 |1 |

20. M =Missed payments

D1= Customer defaults

D2= Customer does not default

P(D1)=0.05

P(D2)=0.95

P(M/D2)=0.2

P(M/D1)=1

[P(D1)*P(M/D1)]/

P(D1/M)=

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