ANSWER ALL QUESTIONS

2.5.1. An unbiased coin is tossed 10 times. What is the probability of getting

(i) exactly 4 heads?;

(ii) exactly 2 tails?;

(iii) first head at the 10th trial?;

(iv) the 3rd head at the 10th trial?

2.5.2. A class has 10 students. Each student has a birthday which can be one of the 365 days of the year, and no other information about the birthdays is available.

(i): A student is selected at random. What is the probability that the student has the birthday on 1 February?

(ii) What is the probability that their birthdays are all distinct, none coinciding with any other?

2.5.3. In a number lottery, each ticket has 3 digits. When the lottery is drawn, a specific sequence of 3 digits will win, the digits could be repeated also. A person has bought 4 tickets. What is the probability that one of his tickets is the winning ticket?

2.5.4. In Exercise 2.5.3, if repetition of the numbers is not allowed, then what is the answer?

2.5.5. From a well-shuffled deck of 52 playing cards, a hand of 8 is drawn at random. What is the probability that the hand contains 4 clubs, 2 spades, 1 heart and 1 diamond?

2.5.6. In a game, an unbiased coin is tossed successively. The game is finished when a head appears. What is the probability that

(i) the game is over with less than or equal to 10 trials;

(ii) the game is over at the 10th trial.

2.5.7. In the same game of tossing an unbiased coin successively, suppose that a person wins the game if a head appears. What is the probability of the person winning the game?

2.5.8. A balanced die is rolled twice. What is the probability of

(i) rolling 6 (sum of the face numbers is 6)?

(ii) getting an even number on both occasions?

(iii) and even number comes in the first trial and odd number comes in the second trial?

2.5.9. In 6/36 lottery, there are 36 numbers and a given collection of 6 will win. A person has 3 such 6/36 tickets. What is the probability that one of these three is the win

2.5.10. In a 7/49 lottery, there are 49 numbers and a specific collection of 7 numbers wins. A person has 3 such tickets.

(i) What is the probability that one of these is the winning ticket (assume that no two tickets will have the same set of numbers);

(ii) Comparing with the probabilities in Exercises 2.5.9

2.5.10 (i), which lottery that a person should prefer 6/36 or 7/49?

2.5.11. A manufacturing unit of water heaters is known to produce 10% of defective items. A customer bought 3 water heaters from this manufacturer. What is the probability that

(i) at least one of the three is defective;

(ii) all are defective?

2.5.12. Vembanad Lake contains n Karimeen (particular fish). A random sample of 50 Karimeen were caught and tagged and then released into the lake. After several months, a random sample of 100 Karimeen were caught.

(i) What is the probability that this sample contains 5 tagged Karimeen?

(ii) How will you estimate n, the total number of Karimeen in the Vembanad Lake based on this information that out of 100 caught 5 were found to be tagged?

2.5.13. A box contains 3 red and 5 green identical balls. Balls are taken at random, one by one, with replacement. What is the probability of getting

(i) 3 red and 5 green in 8 trials;

(ii) a red ball is obtained before a green ball is obtained.

2.5.14. In Exercise 2.5.13, if the balls are taken at random without replacement, what is the probability of getting

(i) the sequence RRGG in four trials;

(ii) RGRR in four trials? (ii) the third ball is green given that the first two were red?

(iii) the third ball is green given that the first ball was red?

(iv) the third ball is green and no other information is available.

2.5.15. Thekkady Wildlife Reserve is visited by people from Kerala, Tamilnadu, Karnataka and from other places. For any day, suppose that the proportions are 50%, 30%, 10%, 10%, respectively. Suppose that the probability that garbage will be thrown around at the reserve, on any day, by visitors from Kerala is 0.9, visitors from Tamilnadu is 0.9, visitors from Karnataka is 0.5 and for others it is 0.10.

(i) What is the probability that the reserve will have garbage thrown around on any given day;

(ii) On a particular day, it was found that the place had garbage thrown around, and what is the probability that it is done by Keralite visitors?

2.5.16. In a production process, two machines are producing the same item. Machine 1 is known to produce 5% defective (items which do not satisfy quality specifications) and Machine 2 is known to produce 2% defective. Sixty percent of the total production per day is by Machine 1 and 40% by Machine 2. An item from the day's production is taken at random and found to be defective. What is the probability that it was produced by Machine 1?

2.5.17. In a multiple choice examination, there are 10 questions and each question is supplied with 4 possible answers of which one is the correct answer to the question. A student, who does not know any of the correct answers, is answering the questions by picking answers at random. What is the probability that the student gets (i) exactly 8 correct answers;

(ii) at least 8 correct answers;

(iii) not more than three correct answers?

2.5.18. There are 4 envelopes addressed to 4 different people. There are 4 letters addressed to the same 4 people. A secretary puts the letters at random to the four envelopes and mails. All letters are delivered. What is the probability that none gets the letter addressed to him/her?

2.5.19. Construct two examples each of practical situations where you have two events A and B in the same sample space such that they are

(i) mutually exclusive and independent;

(ii) mutually exclusive and not independent;

(iii) not mutually exclusive but independent;

(iv) not mutually exclusive and not independent.

2.5.20. For the events A1 ,... ,Ak in the same sample space S, show that

(i) P(A1 A2 Ak ) p(A1 ) + P(A2 ) + + P(Ak );

(ii) P(A1 A2 Ak ) P(A1 ) + + P(Ak ) (k 1).

2.5.21. For two events A and B in the same sample space S, show that if A and B are independent events (that is satisfying the product probability property) then

(i) A and B c ;

(ii) A c and B;

(iii) A c and B c are independent events, where A c and B c denote the complements of A and B in S.