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Downloaded from https:// www.studiestoday.com, , PROBABILITY, , , , Probability, , 24.1 PROBABILITY, , In everyday life, we come across statements such as :, (i) It may rain today., (ii) Probably Rajesh will top the council examination this year., , (iii) I doubt she will pass the test., (iv) It is unlikely that India will win the world cup., (v) Chances are high that the prices of petrol will go up., , The words, ‘may’, ‘probably’, ‘doubt’, ‘unlikely’, ‘chances’ used in the above, statements indicate uncertainties. For example, in the statement (i), ‘may rain today’ means, it may rain or may not rain today. We are predicting rain today based on our past, experience when it rained under similar conditions. Similar predictions are made in other, cases listed (ii) to (v)., , Probability is a measure of uncertainty., , The theory of probability developed as a result of studies of game of chance or, gambling. Suppose you pay @ 5 to throw a die — if it comes up with number 1, you get, Z 20, otherwise you get nothing. Should you play such a game? Does it give a fair chance, of winning? Search for mathematical answers of these types of questions led to the, development of modern theory of probability., , 24.1.1 Some terms related to probabiltiy, , 1. Experiment. An action which results in some (well defined) outcomes is called an, experiment., , 2. Random experiment. An experiment is called random if it has more than one possible, outcome and it is not possible to tell (predict) the outcome in advance., , , , APC pronaaumy |i, , he Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , For example :, (i) tossing a coin, (ii) tossing two coins simultaneously, (iii) throwing a die, (iv) drawing a card from a pack of 52 (playing) cards., All these are random experiments., From now onwards, whenever the word experiment is used it will mean random, , experiment., 3. Sample space. The collection of all possible outcomes of a random experiment is called, , sample space., 4. Event. A subset of the sample space associated with a random experiment is called an event., For example :, (i) When a die is thrown, we can get any number 1, 2, 3, 4, 5, 6. So the sample space, S = {1, 2, 3, 4, 5, 6}. A few events of this experiment could be —, ‘Getting a six’ : {6}, ‘Getting an even number’ : {2, 4, 6}, ‘Getting a prime number’ : {2, 3, 5}, ‘Getting a number less than 5’ : {1, 2, 3, 4}, etc., (11) When a pair of coins is tossed, the sample space of the experiment, S = {HH, HT, TH, TT}. A few events of this experiment could be —, ‘Getting exactly one head’ : {HT, TH}, ‘Getting exactly two heads’ : {HH}, ‘Getting atleast one head’ : {HH, HT, TH}, etc., 5. Occurrence of an event. When the outcome of an experiment satisfies the condition, mentioned in the event, then we say that event has occurred., , For example :, , (i) In the experiment of tossing a coin, an event E may be getting a head. If the coin, comes up with head, then we say that event E has occurred, otherwise, if the coin, comes up with tail, we say that event E has not occurred., , (ii) In the experiment of tossing a pair of coins simultaneously, an event E may be, getting two heads. If the pair of coins come up with two heads, then we say that, event E has occurred, otherwise, we say that event E has not occured., , (i) In the experiment of throwing a die, an event E may be taken as ‘getting an even, number’. If the die comes up with any of the numbers 2, 4 or 6, we say that event, E has occurred, otherwise, if the die comes up with 1, 3 or 5, we say that event, E has not occurred., , 6. Favourable outcomes. The outcomes which ensure the occurrence of an event are called, favourable outcomes to that event. d, , 7. Equally likely outcomes. If there is no reason for any one outcome to occur in preference to, any other outcome, we say that the outcomes are equally likely., , For example :, (i) In tossing a coin, it is equally likely that the coin lands either head up or tail up., , (ii) In throwing a die, each of the six numbers 1, 2, 3, 4, 5, 6 is equally likely to show, , up., , , , (J + UNDERSTANDING ICSE MATHEMATICS - X APC, Downloaded from https:// www.studiestoday.com

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24.1.2 Definition of probability from https:// www.studiestoday.com, , The assumption that all the outcomes are equally likely leads to the following definition, of probability :, , The probability of an event E, written as P(E), is defined as, , P(E) Es number of outcomes favourable to E, total number of possible outcomes of the experiment 7, , , , Sure event. An event which always happens is called a sure event or a certain event., , For example, when we throw a die, then the event ‘getting a number less than 7’ is a, sure event., , The probability of a sure event is 1., , Impossible event. An event which never happens is called an impossible event. For, example, when we throw a die, then the event ‘getting a number greater than 6’ is an, impossible event., , The probability of an impossible event is 0., , Elementary event. An event which has one (favourable) outcome from the sample space is, called an elementary event., , An event which has more than one (favourable) outcome from the sample space is, called a compound event., , For example, when we throw a die, then the event getting number 5 is an elementary, event whereas the event getting an even number (2, 4, or 6) is a compound event., , Complementary events. If E is an event, then the event ‘not E’ is complementary event of E., , For example, when we throw a die, let E be the event getting a number less than or, equal to 2, then the event ‘not E’ i.e. getting a number greater than 2 is complementary, event of E., , Complement of E is denoted by E or E., , Let E be an event, then the number of outcomes favourable to E is greater than, or equal to zero and is less than or equal to total number of outcomes. If follows that, Os P(E) Si., , QO) Let E be an event, then we have :, (i) O< P(E) <1, (ii) P(E) = 1 - P(E), (iii) P(E) = 1 - P(E), (iv) P(E) + P(E) = 1., _ The sum of the probabilities of all the elementary events of an experiment is 1., , ILLUSTRATIVE EXAMPLES, , Example 1. A coin is tossed once. Find the probability of getting, (i) a head (ii) a tail., , Solution. When a coin is tossed once, the possible outcomes are Head (H) and Tail (TT)., So, the total number of possible outcomes = 2., , |, , \ APC PROBABILITY —]], , Downloaded from https:// www.studiestoday.com

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(i) Let E, be the eveh OPRetiing athedamrhencttemuntber of favourable outcomes to, E, (ie. of getting a head) = 1, number of favourable outcomes to E,, , ». P(E,) = =>., , total number of possible outcomes, , (ii) Let E, be the event of getting a tail, then the number of favourable outcomes to, E, (i.e. of getting a tail) = 1., , number of favourable outcomes to E, ee P(E;) = 2 =, , 1, total number of possible outcomes 2°, , , , Note that E, and E, are elementary events and P(E,) + P(E) = ; + a i, , Thus, the sum of the probabilities of all the elementary events of an experiment, = 1,, , Example 2. Malini buys a fish from a shop for her aquarium. The shopkeeper takes out one fish, at random from a tank containing 5 male fish and 8 female fish. What is the probability that the, , fish taken out is a male fish?, , Solution. As a fish is taken out at random from the tank, all the outcomes are equally, likely., , Total number of fish in the tank = 5 + 8 = 13., , .. Total number of possible outcomes = 13., , Let E be the event ‘taking out a male fish’. As there are 5 male fish in the tank, the, number of favourable outcomes to the event E = 5., , -. P(E) = number of favourable outcomes to E —e, G total number of possible outcomes 13, , Example 3. A die is tossed once. Find the probability of getting, (i) number 4 (ii) a number greater than 4, (ti) a number less than 4 (iv) an even number, (v) a number greater than 6 (vi) a number less than 7., , Solution. When a die is tossed once, the possible outcomes are the numbers 1, 2, 3, 4,, 5, 6. So, the total number of possible outcomes = 6., , (i) The event is getting the number 4., The number of favourable outcomes to the event getting number 4 = 1., , “. P (getting number 4) = 7, , (ii) The event is getting a number greater than 4 and the outcomes greater than 4 are, 5 and 6. So, the number of favourable outcomes to the event getting a number, greater than 4 = 2, , 2 1, , . P (getting a number greater than 4) = rae?, , (ili) The event is getting a number less than 4 and the outcomes less than 4 are 1, 2,, , 3. So, the number of favourable outcomes to the event getting a number less than, , 4=3., , . P (getting a number less than 4) = 2. >, , (iv) The event is getting an even number and the even numbers (outcomes) are 2, 4,, 6. So, the number of favourable outcomes to the event getting an even, , number = 3. 1, , .. P (getting an even number) = 3. >., , — «A UNDERSTANDING ICSE MATHEMATICS - X

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(v) The event is gettimpennumbemgneatemthanGiendithenre is no outcome greater than, 6. So, the number of favourable outcomes to the event getting a number greater, than 6 = 0., , .. P (getting a number greater than 6) = : = 0., , Note that in this experiment, getting a number greater than 6 is an impossible, event., , (vi) The event is getting a number less than 7 and all the outcomes 1, 2, 3, 4, 5, 6 are, less than 7. So, the number of favourable outcomes to the event getting a number, less than 7 = 6., , .. P (getting a number less than 7) = : = 1., , Note that in this experiment, getting a number less than 7 is a sure event., , Example 4. A bag contains 5 red, 4 green and 6 white balls. If a ball is drawn at random from, the bag, find the probability that it will be, , (i) white (ii) red (iti) green., , Solution. Saying that a ball is drawn at random from the bag means that all the balls, are equally likely to be drawn., , Total number of balls in the bag = 5 + 4 + 6 = 15., .. The number of possible outcomes = 15., , (i) Let W be the event ‘the ball drawn is white’. As there are 6 white balls in the bag,, the number of favourable outcomes to the event W = 6., , ‘ SI eee, ima) = aaa,, , (ii) Let R be the event ‘the ball draw is red’. As there are 5 red balls in the bag, the, number of favourable outcomes to the event R = 5., , Be ee., “. P(R) = ar or, (iii) Let G be the event ‘the ball drawn is green’. As there are 4 green balls in the bag,, , the number of favourable outcomes to the event G = 4., , , , 4, oie =—., We) S, 2 1 4 6+5+4 15, aS “ee is Te =— = 1, Note that P(W) + P(R)+P(G)= = + 2 + 7 ipeiss 15, , Example 5. If the probability of Sania winning a tennis match is 0-63, what is the probability, of her losing the match?, Solution. Let E be the event ‘Sania winning the match’, then P(E) = 0:63 (given)., .. P (Sania losing the match) = P (not E) = P(E), = 1- P(E) = 1-063, =:0:37., Note that winning the match and losing the match are complementary events., , Example 6. Ankita and Nagma are friends. They were both born in 1990. What is the, probability that they have, , (i) same birthday? (ii) different birthdays?, , Solution. Out of two friends, say, Ankita’s birthday can be any day of the 365 days in, a (non-leap) year. Also Nagma’s birthday can be any day of the 365 days of the same year., So, the total number of outcomes = 365., , ; APC PROBABILITY 4/4, ., , Downloaded from https:// www.studiestoday.com