# PROBABILITY. The sample space of the experiment of tossing two coins is given by

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1 PROBABILITY Introduction Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability of an event is always a number between 0 and 1 both 0 and 1 inclusive. If an event s probability is nearer to 1, the higher is the likelihood that the event will occur; the closer the event s probability to 0, the smaller is the likelihood that the event will occur. If the event cannot occur, its probability is 0. If it must occur (i.e., its occurrence is certain), its probability is 1. Random experiment An experiment is random means that the experiment has more than one possible outcome and it is not possible to predict with certainty which outcome that will be. For instance, in an experiment of tossing an ordinary coin, it can be predicted with certainty that the coin will land either heads up or tails up, but it is not known for sure whether heads or tails will occur. If a die is thrown once, any of the six numbers, i.e., 1, 2, 3, 4, 5, 6 may turn up, not sure which number will come up. (i) Outcome A possible result of a random experiment is called its outcome for example if the experiment consists of tossing a coin twice, some of the outcomes are HH, HT etc. (ii) Sample Space A sample space is the set of all possible outcomes of an experiment. In fact, it is the universal set S pertinent to a given experiment. A. The sample space for the experiment of tossing a coin twice is given by S = {HH, HT, TH, TT} B. The sample space for the experiment of drawing a card out of a deck is the set of all cards in the deck. Event: An event is a subset of a sample space S. For example, the event of drawing an ace from a deck is A = {Ace of Heart, Ace of Club, Ace of Diamond, Ace of Spade} Types of events (i) Impossible and Sure Events The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called a sure event. (ii) Simple or Elementary Event If an event E has only one sample point of a sample space, i.e., a single outcome of an experiment, it is called a simpler elementary event. The sample space of the experiment of tossing two coins is given by S = {HH, HT, TH, TT} The event E1 = {HH} containing a single outcome HH of the sample space S is a simple or elementary event. If one card is drawn from a well shuffled deck, any particular card drawn like queen of Hearts is an elementary event.

2 (iii) Compound Event If an event has more than one sample point it is called a compound event, for example, S = {HH, HT} is a compound event. (iv) Complementary event Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A. The complement of A is denoted by A or A. It is also called the event not A. Further P(A) denotes the probability that A will not occur. A = A= S A = {w: w S and w A} Event A or B : If A and B are two events associated with same sample space, then the event A or B is same as the event A B and contains all those elements which are either in A or in B or in both. Further more, P (A B) denotes the probability that A or B (or both) will occur. Event A and B If A and B are two events associated with a sample space, then the event A and B is same as the event A B and contains all those elements which are common to both A and B. Further more, P (A B) denotes the probability that both A and B will simultaneously occur. The Event A but not B (Difference A B) An event A B is the set of all those elements of the same space S which are in A but not in B, i.e., A B = A B. Mutually exclusive : Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously, and thus P(A B) = 0. Remark Simple or elementary events of a sample space are always mutually exclusive. For example, the elementary events {1}, {2}, {3}, {4}, {5} or {6} of the experiment of throwing a dice are mutually exclusive. Consider the experiment of throwing a die once. The events E = getting a even number and F = getting an odd number are mutually exclusive events because E F = φ Note For a given sample space, there may be two or more mutually exclusive events. Exhaustive events If E1, E2,..., En are n events of a sample space S then E 1, E2,..., En are called exhaustive events. In other words, events E1, E2,..., En of a sample space S are said to be exhaustive if atleast one of them necessarily occur whenever the experiment is performed. Consider the example of rolling a die. We have S = {1, 2, 3, 4, 5, 6}. Define the two events A : a number less than or equal to 4 appears. B : a number greater than or equal to 4 appears.

3 Now A : {1, 2, 3, 4}, B = {4, 5, 6} A B = {1, 2, 3, 4, 5, 6} = S Such events A and B are called exhaustive events. Mutually exclusive and exhaustive events If E1, E2,..., En are n events of a sample space S and if i E j = φfor every i j, i.e., Ei and Ej are pairwise disjoint and 1ESn i= = U, then the events E1, E2,... En are called mutually exclusive and exhaustive events. Consider the example of rolling a die. We have S = {1, 2, 3, 4, 5, 6} Let us define the three events as A = a number which is a perfect square B = a prime number C = a number which is greater than or equal to 6 Now A = {1, 4}, B = {2, 3, 5}, C = {6} Note that A B C = {1, 2, 3, 4, 5, 6} = S. Therefore, A, B and C are exhaustive events. Also A B = B C = C A = φ Hence, the events are pairwise disjoint and thus mutually exclusive. Classical approach is useful, when all the outcomes of the experiment are equally likely. We can use logic to assign probabilities. To understand the classical method consider the experiment of tossing a fair coin. Here, there are two equally likely outcomes - head (H) and tail (T). When the elementary outcomes are taken as equally likely, we have a uniform probablity model. If there are k elementary outcomes in S, each is assigned the probability of 1/k. Therefore, logic suggests that the probability of observing a head, denoted by P (H), is 1/2= 0.5, and that the probability of observing a tail, denoted P (T), is also 1/2=.5. Notice that each probability is between 0 and 1. Further H and T are all the outcomes of the experiment and P (H) + P (T) = 1. Classical definition If all of the outcomes of a sample space are equally likely, then the probability that an event will occur is equal to the ratio : 1. The number of outcomes favourable to the event 2. The total number of outcomes of the sample space 3. Suppose that an event E can happen in h ways out of a total of n possible equally likely ways.

4 Then the classical probability of occurrence of the event is denoted by P (E) = h/n The probability of non occurrence of the event E is denoted by P (not E) = h/n-h =1-P(E) Thus P (E) + P (not E) = 1 The event not E is denoted by E or E (complement of E) Therefore P ( E) = 1 P (E ) Axiomatic approach to probability Let S be the sample space of a random experiment. The probability P is a real valued function whose domain is the power set of S, i.e., P (S) and range is the interval [0, 1] i.e. P : P (S) [0, 1] satisfying the following axioms. (i) For any event E, P (E) 0. (ii) P (S) = 1 (iii) If E and F are mutually exclusive events, then P (E F) = P (E) + P (F). It follows from (iii) that P ( φ) = 0. Let S be a sample space containing elementary outcomes w1, w2,..., wn,i.e., S = {w1, w2,..., wn} It follows from the axiomatic definition of probability that (i) 0 P (w i) 1 for each w i S (ii) P (w i) + P (w2) P (w n) = 1 (iii) P (A) = P( ) i w for any event A containing elementary events w i. For example, if a fair coin is tossed once P (H) = P (T) = ½ satisfies the three axioms of probability. Now suppose the coin is not fair and has double the chances of falling heads up as compared to the tails, then P (H) = ½ 3 and P (T) = 1/3. This assignment of probabilities are also valid for H and T as these satisfy the axiomatic definitions. Probabilities of equally likely outcomes Let a sample space of an experiment be S = {w1, w2,..., wn} and suppose that all the outcomes are equally likely to occur i.e.,the chance of occurrence of each simple event must be the same i.e., P (wi) = p for all wi S, where 0 p 1 Since 1P( ) 1niiw== i.e., p+p+ p p(ntimes) = 1 n p= 1, i.e. p= 1n Let S be the sample space and E be an event, such that n(s) = nand n(e) = m. If each outcome is equally likely, then it follows that

5 P (E) = m/n = Number of outcomes favorable to E /Total number of possible outcomes Addition rule of probability If A and B are any two events in a sample space S, then the probability that atleast one of the events A or B will occur is given by P (A B) = P (A) + P (B) P (A B) Similarly, for three events A, B and C, we have P (A B C) = P (A) + P (B) + P (C) P (A B) P (A C) P (B C) + P (A B C) Addition rule for mutually exclusive events If A and B are disjoint sets, then P (A B) = P(A) + P (B) [since P(A B) = P(φ) = 0, where A and B are disjoint]. The addition rule for mutually exclusive events can be extended to more than two events. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the condition that the event F has occurred, written as P (E F), is given by missing Properties of Conditional Probability Let E and F be events associated with the sample space S of an experiment. Then: (i) P (S F) = P (F F) = 1 (ii) P [(A B) F] = P (A F) + P (B F) P [(A B F)], where A and B are any two events associated with S. (iii) P (E F) = 1 P (E F) Multiplication Theorem on Probability Let E and F be two events associated with a sample space of an experiment. Then = P (E F) = P (E) P (F E), P (E) 0 = P (F) P (E F), P (F) 0 If E, F and G are three events associated with a sample space, then P (E F G) = P (E) P (F E) P (G E F)

6 1. If E and F are two events associated with a random experiment for which P (E ) = 0.60, P (E or F ) = 0.85, P (E and F ) = 0.42, find P(F).[Delhi 1998C] (0.67) 2. If A and B are two events such that P(A) = 0.5, P(B) = 0.6 and P(A B) = 0.8. Find P(A/B) and P(B/A). 3. If A and B are two events such that P(A) = 3/8, P(B) = 5/8 and P(B/A) = 2/3, find P(A/B) & P(A B). 4. Let A & B be events with P(A B) = 7/8, P(A B) = 1/4 and P (A) = 5/8 find P(A), P(B), P(A B). Let A and B be events with P(A) = 1/a, P(A u B) = and P (B) = 5/8. Find 5. (i) P(An B) (ii) P(A n B) (iii) P(A u B) (iv) P(B n A) 6. Let A and B be events with P(A) = 1/3 and P(B) = and P(A u B) =.Find 7. (i) P(A/B) (ii) P(B/A) (iii) P(A n B) (iv) P(A/B) 8. Let S={ a, b, c, d, e, f}.given P(a) = ; P(b) =1/16 ; P(c) = 1/8 ; P(d) =3/16 ; P(e) =, & P(f) = 5/16 9. LetA={a,c,e} B = {c,d,e,f} C = {b,c,f} a) Find (i) P(A/B) (ii) P(B/C) (iii) P(C/A) (iv) P(A/C) b) Prove that (i) P(E/E) = 1 (ii) P(F/E) = 0 if EnF = c) (ii)ecf=>p(f/e) = 1 (iv)ifej ce2thenp(ei/f)<p(e2/f). 10. A coin is tossed twice and the four possible outcomes are assumed to be equally likely. If E is the event: " both heads and tails have occurred " and F the event: " atmost one tail is observed ". Find F(E), P(F), P(E/F), P(F/E). 11. ln a class, 25% of the students failed in mathematics, 15% of the students failed in chemistry and 10% of the students failed both in maths and chemistry. A student is selected at random. a) If he failed in chemistry, what is the probability that he failed in mathematics? b) If he failed in maths, what is the probability that he failed in chemistry? '1% c) What is the probability that he failed in mathematics or chemistry? 12. A die is rolled. If the outcome is an odd number, what is the probability that it is prime? 13. A die is rolled. If the outcome is an even number, what is the probability that it is a number greater ' than Two dice are thrown. Find the probability that the numbers appeared have a sum 8, if it is known that the second die always exhibits A pair of fair dice is thrown. Find/the probability that the sum is 8 or greater if 5 appears on the first die. 16. A pair of dice is thrown. If tyro numbers appearing are different, find the probability that the sum is 4 or less. 17. A couple has 2 children. Find the probability that both are toys, if it is known that (i) one of the children is a boy (ii) the older child is a boy. 18. Three coins are tossed. Find the probability that they are all heads, if one of the coin shows a head. Two numbers are selected at random from the integers 1 through 9. If the sum is even, find the probability that both numbers are odd. 19. Find P(B/A) if (i) A is a subset of B. (ii) A and B are mutually exclusive. 20. A bag contains 3 red and 4 black balls and another bag has 4 red and 2 black balls. One bag is selected, each of the two bags is equally likely to be selected. From the selected bag a ball is drawn, each ball in the bag is equally likely to be drawn. 21. Let E be the event: "the first bag is selected" F be the event: "the IInd bag is selected" G be the events the ball drawn is red" Find P(E), P(F), P(G/E), P(G/F), P(G)

8 (a) P (F E) = P (F), provided P (E) 0 (b) P (E F) = P (E), provided P (F) 0 Using the multiplication theorem on probability, we have (c) P (E F) = P (E) P (F) Three events A, B and C are said to be mutually independent if all the following conditions hold: a) P (A B) = P (A) P (B) b) P (A C) = P (A) P (C) c) P (B C) = P (B) P (C) d) P (A B C) = P (A) P (B) P (C) Independent invents 1) Let A and B be events with P(A) = ¼, P(A B ) = 1/3 and P(B) = X. a) Find x if A and B are mutually exclusive. (1/12) b) Find x if A and B are independent. (1/9) c) Find x if A is a subset of B. (1/3) 2) If A and B are two independent events such that P(A)= 0.65, P(A u B) = 0.65 and P(B) = p. Find the value of p. (6/13) 3) If The probability of a student A passing on examination is 3/5 and of student B passing is 4/5. Assuming the two events: "A passes", "B passes" are independent, find the probability of a) both students passing the examination. (12/25) b) only A passing the examination. (3/25) c) only B passing the examination. (8/25) d) only one of them passing the examination. (11/25) e) at least one of A and B will pass the examination. (23/25) f) none of them passing the examination. (2/25) 4) Let A = event that a family p has children both male and female, and let B = event that a family has atmost one boy. a) Show that A and B are independent events if a family has three children. (BBB,BBG,BGB,BGG,GBB,GBG,GGB,GGG ) b) Show that A and B are dependent events if a family has two children. 5) A pair of fair coins are tossed and all the four outcomes are assumed to be equally likely. Let events A = "heads on the first coins" B = "heads on the second coin" C = "heads on exactly one coin" Show that the three events are not independent. 6) An unbiased die is tossed twice. Find the probability of getting a 4,5 or 6 on the first toss and a 1,2,3 or 4 on the II nd toss. (1/3)

9 7) Wife speaks truth in 60% of the cases and husband in 90% of the cases. what percentage of cases are they likely to contradict each other in stating the same fact. (42%) 8) The bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that a) both are red (9/40) b) both are black () c) one is red and one is black (21/40) 9) Two cards are drawn at random from a well shuffled pack of 52 cards. What is the probability that X (a) both are aces (1/221) (b) both are red (25/102) (c) at least one is an ace. (33/221). 10) A bag contains 3 red and 2 black balls. One ball is drawn from it at random. It's color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing: (i) two red balls (9/25) (ii) two black balls. (4/25) (iii) one red and one black ball. (12/25) 11) Three cards are drawn from a well shuffled deck of cards, one after the another and with replacement. What is the probability that: (a) all the three cards are spade. (1/64) (b) first two cards are queens & third card is a black card. (1/338) (c) first card is jack, second is red queen and third card is a king. (1/4394) 12) A card is drawn from a well shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck is reshuffled. Another card is drawn from the deck, what is the probability that (a) both the cards of the same suit. (%) (b) both cards are ace. (1/169) (c) the first card is a spade and the second card is a black king. (1/104) (d) both are face cards. (16/169) 13) A die is thrown thrice, find the probability of (i) not getting 5 in any throw. (125/216) (ii) getting 5 atleast once in the three throws. (91/216) 14) What is the probability of getting all the heads in four throws of a coin? (1/16) 15) problem of probability is given to three students A,B,C whose chances of solving it are 1/4,1/3 and 1/5 'respectively. What is the probability that the problem will be solved? 16) Three selected for interview for 3 posts. For the first post there are 5 candidates, for the second there 4 and for the third there are 6. If the selection of each candidate is equally likely, find the chance that Vinita will be selected for atleast one post. 17) The probabilities of A,B,C solving a problem are 1/3,2/7 and 3/8 respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them will solve it. (25/56) 18) Three critics review a book. Odds in favour of the book are 5:2; 4:3 and 3:4 respectively for the three critics. Find the probability that majority are in favour of the book. (209/343) 19) Three groups of children contains 3 girls and 1 boy, 2 girls and 2 boys, 1 girl and 3 boys respectively. One child is selected at random from each group. Find the chance that the three selected comprise 1 girl and 2 boys. (13/32)

10 20) A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black? (5/22) 21) A bag contains 8 red and 6 green balls. Three balls are drawn one by one without replacement. Find the probability that atleast two balls drawn are green. (5/13) 22) An urn contains 5 white and 8 black balls. Two successive drawing of three balls at a time are made such that: a) the balls are replaced before the second trial. (140 / 20449) b) the balls are not replaced before the second trial. (7 / 429) c) Find the probability in each case that the 1 st drawing will give 3 white and the second 3 blaek balls. 23) Four cards are drawn from a pack of 52 cards without replacement. What is the probability that they are of all of different suits? (2197/20825) 24) 1 black and 2 white balls, in another there are 2 black and 1 white balls. A ball is drawn from the first and put into the second and then a ball is drawn from the second urn. Show that the chance that it is a white is 5/12. 25) Three persons A,B,C throw a die in succession till one get a six and wins the game. Find their respective probabilities of winning. (36/91] 30/91,25/91) 26) Calculate the probability obtaining a six atleast once by throwing a die four times. (l-(5/6) 4 ) 27) Calculate the probability of obtaining a double six atleast once by throwing two dice twenty four times. " (l-(35 / 36) 24 ) 28) A and B take turn in throwing two dice, the first to throw 9 being awarded. Show that if A has the first throw, their chances of winning are in the ratio 9: 8. Partition of a Sample Space A set of events E1, E2,..., En is said to represent a partition of a sample space S if (a) Ei Ej = φ, i j; i, j= 1, 2, 3,..., n (b) Ei E2... En= S, and (c) Each Ei φ, i. e, P (Ei) > 0 for all i= 1, 2,..., n Theorem of Total Probability Let {E1, E,..., En} be a partition of the sample space S. Let A be any event associated with S, then P (A) = P(E i ) P(A E i ) Bayes Theorem If E1, E2,..., En are mutually exclusive and exhaustive events associated with a sample space, and A is any event of non zero probability, then P(E i A) = P(E i )P(A E i ) P(E i )P(A E i )

12 10. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die? (Ncert) 11. A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A? (Ncert) 12. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond. (Ncert) 13. In a bolt factory, machines A,B and C manufacture respectively 25%, 35% and 40% of the total bolts. Of dieir outputs 5%, 4% and 2% are respectively defective bolts. A bolt is drawn at random from the product. a) what is the probability that the bolt drawn is defective? b) if the bolt drawn is found to be defective, what is the probability that it is manufactured by the "machine A or C? (Ncert) 14. A bag X contains 2 white and 3 red balls and a bag Y contains 4 white and 5 red balls. One ball is drawn at random from one of the bags & is found to be red. Find the probability that it was drawn from bag Y. 15. Three urn contains 6 red, 4 black; 4 red, 6 black and 5 red and 5 black respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is from the first urn. 16. A company has two plants to manufacture scooter. Plant I manufactures 70% of scooters and part II manufactures 30%. At plant I, 80% of the scooters are rated as of standard quality and at plant II, 90% of the scooters are rated a of standard quality. A scooter is chosen at random and is found to be standard quality. What is the probability that it has come from part II? 17.The contents of urn I, II, III are as follows: Urn I : 1 white, 2 black and 3 red balls. Urn II : 2 white, 1 black and 1 red ball. Urn III: 4 white, 5 black and 3 red balls. 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, III? 18. There are 3 bags, each containing 5 white balls and 3 black balls. Also, there are 2 bags, each containing 2 white balls and 4 black balls. A white ball is drawn at random. Find the probability that this white ball is from a bag of the first group?

13 19. A letter has known to come either from Calcutta or Tatanagar if TA is visible on the post office stamp. What is the probability that it has come from Tatanagar? (Ans: 7/ ) 20. A card from a pack of 52 cards is lost. From the remaining cards of the pack, 2 cards are drawn and are found to be diamonds. Find the probability of the missing card to be a diamond card? (Ans: 11/ 50) 21. A pair of dice is-thrown twice. If the random variable X is defined as the number of doublets, find the probability distribution of X. (2006) 22. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.( Example 16) 23. Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?( Example 17) 24. Suppose that the reliability of a HIV test is specified as follows: Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the test are judged HIV ive but 1% are diagnosed as showing HIV+ive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV+ive. What is the probability that the person actually has HIV?( Example 18) 25. In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?( example 19) 26. Example 20 A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively, and. The probabilities that he will be late are,,and,if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he comes by train? 27. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. Example A company has two plants to manufacture bicycles. The first plant manufactures 60% of the bicycles and the second plant 40%. 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked up at random and found to be of standard quality. Find the probability that it comes from the second plant. [Delhi 2003] 29. A factory has three plants A, B and C. Their daily production is 500,1000 and 2000 units. Out of these 0.5%, 0.8% and 1% units respectively are found to be defective. An item is chosen at random and is found to be defective. What is the probability that it came from plant A? [Foreign 2003] 30. A company has two plants to manufacture scooters. Plant I manufactures 70% of the scooters and plant II manufactures 30%. At plant I, 30% of the scooters are rated of standard quality and at plant II, 90% of the scooters are rated of standard quality. A scooter is chosen at random and is found to be of standard quality. Find the probability that it has come from plant II. [Delhi 2000] 27/83

14 31. Random Variable and its Probability Distribution A random variable is a real valued function whose domain is the sample space of a random experiment. The probability distribution of a random variable X is the system of numbers X : x1 x2... xn P(X) : p1 p2... pn where pi> 0, i=1, 2,..., n, pi=1. Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of the number of aces.[ Example 24] Mean and Variance of a Random Variable Let X be a random variable assuming values x1, x2,..., xn with probabilities p1, p2,..., pn, respectively such that pi 0, pi=1, Mean of X, denoted by μ [or expected value of X denoted by E (X)] is defined as μ= E (X) = pixi and variance, denoted by σ2, is defined as var(x) = E(X2) [E(X)]2 where [E(X)]2 = xi2 p(xi) Bernoulli Trials Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions: (i) There should be a finite number of trials (ii) The trials should be independent (iii) Each trial has exactly two outcomes: success or failure (iv) The probability of success (or failure) remains the same in each trial. Binomial Distribution A random variable X taking values 0, 1, 2,..., n is said to have a binomial distribution with parameters nand p, if its probability distibution is given by

15 P (X = r) =ncr pxqn-x = n! x!(n x)! px qn-x where q = 1 p and r = 0, 1, 2,..., n Solved Examples 1. A coin is tossed twice. Find the possibility of A. getting two heads 1/4 B. getting no heads 1/4 C. getting at least 1 head 3/4 D. getting l head & 1 tail ½ 2. Three coins are tossed. Find the probability of 3. getting exactly 2 heads (i) 3/8 4. getting atleast 2 heads 1/2 5. getting atleast 1 head and 1 tail 3/4 6. Two unbiased dice are thrown. Find the probability that 7. The first dice shows 6. (1/6) 8. Both the dice show that same number (getting a doublet). (1/6) 9. The total of the numbers on the dice is 8. (5/36,) 10. The total of the numbers on the dice is greater than 8. 5/ The total of the numbers on the dice is 13. (0) 12. In a single throw of two dice (a) find the probability of not getting the same number on both the dice, (b) the total of numbers on the two dice is at least 4. (2006) (11/12 ) 13. Two dice are thrown. Find the probability that a multiple of 2 occurs on one dice and a multiple of 3 occurs on the other. (11/36) 14. What are the odds in favour of throwing atleast 7 in a single throw with two dice? (7/5) 15. In a throw of 3 dice, find the probability of getting a sum of more than 14? (5/54) 16. Three dice are thrown together. Find the probability of getting (a) a total of 5 (b) a total of atmost A card is drawn from a pack of cards. Find the probability that it is a) a black card 1/2 b) a red card 1/2 c) a club 1/4 d) an ace (1/13) e) a red king (1/26) f) ace of spades (1/52) g) not a spade 3/4 h) a king or a queen (2/13) i) a red or a king (7/13) 18. Four cards are drawn at random from a pack of 52 cards. What is the probability of getting all the four of same number.

16 19. Four cards are drawn at random from a pack of 52 cards. What is the probability of getting 3 diamonds and one spade. 20. A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that: - (i) 1 is red and 2 are white (ii) 2 are blue and 1 is red (iii) none is red (iv) atleast 1 is red. 21. From a group of 3 boys and 2 girls, two children are selected at random. Find the probability atleast one girl is selected. (7/10) 22. A letter of English alphabet is chosen at random. Calculate the probability that the letter so chosen a) is a vowel (5/26) b) precedes K and is a vowel (3/26) c) follow K and is a vowel (2/26) 23. In a lottery of 50 tickets numbered 1 to 50, two tickets are drawn simultaneously, find the probability that :- a) both the tickets drawn have prime numbers b) none of the tickets drawn has prime numbers. (i) 15/50 (ii) 35/ An integer is chosen at random from the first two hundred digits. What is the probability that the integer chosen is divisible by 6 or A box contains 15 electric bulbs, out of which 2 are defective. Two bulbs are chosen at random from the box. What is the probability that atleast one of these is defective. (9/35) 26. Tickets numbered from 1 to 24 are mixed up together and then a ticket is drawn at random a) What is the probability of drawing a number exactly divisible by 3? (1/3) b) What are the odds against drawing a number exactly divisible by 4? (3/1) 27. A card is drawn from an ordinary deck and gambler bets that it is a spade or an ace. What are the odds against his winning the bet? 9/4 28. The letters of words 'SOCIETY' are placed in random in a row. What is the probability that three vowels come together? (1/7) 29. E, F and G are three events associated with the sample space S of a random experiment. If E, F and G also denote the subsets of S representing these events, what are the sets representing the events, (a) out of the three events atleast two events occur (b) out of the three events only E occurs, (c) out of the three events only one occurs. (d) out of the three events not more than two occur, (e) out of the three events exactly two events occur. (f) none of the three events occur. 30. From a class of 10 boys and 8 girls, a group of 3 students are selected in such a manner that every group of 3 students is equally likely to be selected. Find the probability that the selected group has (i) all boys (ii) all girls (iii) 2 boys and 1 girl. 31. A clerk was asked to mail three report cards to three students. He addresses three envelopes but unfortunately paid no attention to which report card be put in which envelope. What is the probability that exactly one of the students received his or her report card.

17 32. A combination lock on a suitcase has 3 wheels each labeled with nine digits from 1 to 9. if an opening combination is a particular sequence of three digits with no repeats, what is the probability of a person guessing the right combination? (l/504) 33. A did has two faces each with number '1', three faces each with number '2' & one face with number '3'. If the die is rolled once, determine (i) P(2) (ii) P(1 or 3) (iii) P(not 3). (3/6, 3/6, 5/6). ALGEBRA OF EVENTS' 1. Two cards are drawn at random from a well-shuffled pack of 52 cards. What is the probability that either both are red or both are kings? (55/221) 2. Four cards are drawn from a full pack of cards. Find the probability that at least one of the four cards is. 3. In the urn there are 4 white and 4 black balls. What is the probability of drawing the first ball white, the second black, the third white and fourth black and so on. (1/70) 4. A bag contains 2 white marbles, 4 blue marbles and 6 red marbles. Three marbles are drawn from the bag. What is the probability that (a) they are all blue (1/55) (b) they are all red (1/11) (c) they are all ace. 5. Five men in a company of 20 are graduates. If 3 men are picked out of 20 at random, what is the probability that they all are graduates? What is the probability of atleast one graduate? (1/114,137/228) 6. If from a lottery of 30 tickets, marked 1,2, 3,..., 30 four tickets are to be drawn, what is the chance that those marked 1 and 2 are among them? (2/145) 7. If four coins are-tossed, find the chance thatjiere should be two tails? (3/8) 8. If 5 coins are tossed, what is the probability that 2 heads and 3 tails will come up? (5/16) 9. Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur simultaneously is Find the probability that neither A nor B occurs. (0.39) 10. A die is thrown-twice. What is the probability that atleast one of the two numbers is 4. (11/36) 11. In a single throw of two dice, find the probability that neither a doublet (the same number on both sides) nor a total of 9 will appear. (13/18) 12. Two dice are tossed once. Find the probability of getting an odd number first die or a total of 8. (7/12) 13. If the probability of a horse A winning a race is 1/5 and the probability of the hope B winning the same race is A, what is the probability that one of the horse will win? (9/20) 14. From a well-shuffled deck of 52 cards, 4 cards are drawn at random. What is the probability that all the cards are of the same color? (92/833) 15. Four cards are drawn at a time from a pack of 52 playing cards. Find the probability of getting all the four cards of the same suit. (44/4165 ) 16. A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.(28/52) 17. A lady buys a dozen eggs of which two turn out to be bad. She chooses 4 eggs to scramble for breakfast. Find the chance that she chooses (a) all good eggs (14/33) (b) three good and one bad (16/33) (c) at least one bad egg (19/33). 18. In a class 40% of students offered mathematics, 30% offered physics and 20% offered both. If a student is selected at random, find the probability that he has offered mathematics or physics?

18 19. A drawer contains 50 bolts and 150 nutts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or a bolt. (5/8) 20. A box contains 25 tickets numbered 1 to 25. Two tickets are drawn at random. What is the probability that the products of numbers is even? (37/50) 21. There are three events A, B and C, one of which must and only one can happen; the odds are 7 to 3 against A and 6 to 4 against B. Find the odds against C. (7/3) 22. Only one of the three events A, B, C can happen and one of which is must. Given that the chance of A is one third of B and the odd are 2: 1 against C, find the odds in favour of A. (1/5) 23. In a given race; the odds in favour of four horses A, B, C, D are 1: 3, 1: 4,1: 5,1: 6 respectively, find the chance that one of them wins the race. (319/420) 24. A class contains 10 men and 20 women of which half the men and half the women have brown eyes. Find the probability p that a person chosen at random is a man or has brown eyes. (2/3) 25. A basket contains 20 apples and 10 organes, out of which 5 apples and 3 organes are rotton. If a person picks up twcrfruits from the basket at random, find the probability that either both are apples or both are good. (316/435) 26. Find the probability of almost two tails or atleast two heads in a toss of 3 coins. 27. X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting Grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets (i) Grade A in all subjects (ii) Grade A in no subject (iii) Grade A in two subjects (OD 2005) 28. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red (ii) the first ball is black and the second is red (iii) one of them is black and the other red. (OD 2005) In a town of 6000 people 1200 are over 50 years old and 2000 are female. It is known that 30% of the females are over 50 years. What is the probability that a random chosen individual from the town is either female or over 50 years? (Imprtant) (13/30) 30. What is the probability that in a group of (i) 2 people, both will have the same birthday? (ii) 3 people, at least 2 will have the same birthday? Assuming that there are 365 days in a year and on one has his/her birthday on 29 th February. (Imprtant) (1/365, 1/(365) 31. The probability that a person will get an electric contract is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting at least one contract is 2/3, what is the probability that he will get both? (Imprtant) (17/105)

19 Random Variables and its Probability Distributions 1. Example 22 A person plays a game of tossing a coin thrice. For each head, he is given Rs 2 by the organiser of the game and for each tail, he has to give Rs 1.50 to the organiser. Let X denote the amount gained or lost by the person. Show that X is a random variable and exhibit it as a function on the sample space of the experiment. 2. Example 23 A bag contains 2 white and 1 red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If X denotes the number of red balls recorded in the two draws, describe X. 3. Probability distribution of a random variable Example 24Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of the number of aces. Example 25Find the probability distribution of number of doublets in three throws of a pair of dice. Example 26 Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values x, has the following form, where k is some unknown constant (a) Find the value of k. (b) What is the probability that you study at least two hours? Exactly two hours? At most two hours? 1) Find the probability distribution of number of heads in three tosses of a coin. [Delhi 1998] 2) Two cards are drawn successively, with replacement, from a pack of 52 cards. Find the probability distribution of the number of aces drawn.[ai 1998 ; Foreign 2006]

20 Mean of a random variable Example 27Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X. Variance of a random variable Example 28 Find the variance of the number obtained on a throw of an unbiased die. Example 29 Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings. An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable? 3. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X? 4. Find the probability distribution of (i) number of heads in two tosses of a coin. (ii) number of tails in the simultaneous tosses of three coins. (iii) number of heads in four tosses of a coin. 5. Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as (i) number greater than 4 (ii) six appears on at least one die 6. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs. 7. A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails. 8. A random variable X has the following probability distribution:

21 9.The random variable X has a probability distribution P(X) of the following form, where kis some number : (a) Determine the value of k. (b) Find P (X < 2), P (X 2), P(X 2). 10. Find the mean number of heads in three tosses of a fair coin. 11. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X. 12. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X). 13. Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. 14. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X. 15. In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X). Bernoulli Trials and Binomial Distribution Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions : (i) There should be a finite number of trials. (ii) The trials should be independent.

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