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PROBABILITY 387, , (ii) What is the sample space if we are interested in the number of girls in the, family?, , 9. Abox contains | red and 3 identical white balls. Two balls are drawn at random, , in succession without replacement. Write the sample space for this experiment., , 10. An experiment consists of tossing a coin and then throwing it second time if a, head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the, sample space., , 11. Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and, classified as defective (D) or non — defective(N). Write the sample space of this, experiment., , 12. Accoin is tossed. If the out come is a head, a die is thrown. If the die shows up, an even number, the die is thrown again. What is the sample space for the, experiment?, , 13. The numbers 1, 2, 3 and 4 are written separatly on four slips of paper. The slips, are put in a box and mixed thoroughly. A person draws two slips from the box,, one after the other, without replacement. Describe the sample space for the, experiment., , 14. Anexperiment consists of rolling a die and then tossing a coin once if the number, on the die is even. If the number on the die is odd, the coin is tossed twice. Write, the sample space for this experiment., , 15. Acoinis tossed. If it shows a tail, we draw a ball from a box which contains 2 red, and 3 black balls. If it shows head, we throw a die. Find the sample space for this, experiment., , 16. A die is thrown repeatedly untill a six comes up. What is the sample space for, this experiment?, , 16.3, Event, , We have studied about random experiment and sample space associated with an, experiment. The sample space serves as an universal set for all questions concerned, with the experiment., , Consider the experiment of tossing a coin two times. An associated sample space, is S = {HH, HT, TH, TT}., , Now suppose that we are interested in those outcomes which correspond to the, occurrence of exactly one head. We find that HT and TH are the only elements of S, corresponding to the occurrence of this happening (event). These two elements form, the set E= { HT, TH}, , We know that the set E is a subset of the sample space S . Similarly, we find the, following correspondence between events and subsets of S., , 2021-22

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388 MATHEMATICS, , Description of events Corresponding subset of *S’, Number of tails is exactly 2 A= {TT}, , Number of tails is atleast one B = {HT, TH, TT}, , Number of heads is atmost one C = {HT, TH, TT}, , Second toss is not head D= { HT, TT}, , Number of tails is atmost two S = (HH, HI. TH, TT}, Number of tails is more than two o, , The above discussion suggests that a subset of sample space is associated with, an event and an event is associated with a subset of sample space. In the light of this, we define an event as follows., , Definition Any subset E of a sample space S is called an event., , 16.3.1 Occurrence of an event Consider the experiment of throwing a die. Let E, denotes the event “a number less than 4 appears”. If actually ‘1’ had appeared on the, die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,, we say that event E has occurred, , Thus, the event E of a sample space S is said to have occurred if the outcome, of the experiment is such that @ € E. If the outcome @ is such that m ¢ E, we say, that the event E has not occurred., , 16.3.2 Types of events Events can be classified into various types on the basis of the, elements they have., , 1. Impossible and Sure Events The empty set » and the sample space S describe, events. In fact is called an impossible event and S, i.e., the whole sample space is, called the sure event., , To understand these let us consider the experiment of rolling a die. The associated, sample space is, , S= {1, 2,3, 4,5, 6}, , Let E be the event “ the number appears on the die is a multiple of 7”. Can you, write the subset associated with the event E?, , Clearly no outcome satisfies the condition given in the event, i.e., no element of, the sample space ensures the occurrence of the event E. Thus, we say that the empty, set only correspond to the event E. In other words we can say that it is impossible to, have a multiple of 7 on the upper face of the die. Thus, the event E= @ is an impossible, event., , Now let us take up another event F “the number turns up is odd or even”. Clearly, , 2021-22

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PROBABILITY 389, , F={1,2,3,4,5,6,} =S, i.e., all outcomes of the experiment ensure the occurrence of, the event F. Thus, the event F = S is a sure event., , 2. Simple Event If an event E has only one sample point of a sample space, it is, called a simple (or elementary) event., In a sample space containing n distinct elements, there are exactly n simple, events., For example in the experiment of tossing two coins, a sample space is, S={HH, HT, TH, TT}, There are four simple events corresponding to this sample space. These are, E.= {HH}, E,={HT}, E,= { TH} and E.={TT}., 3. Compound Eyent If an event has more than one sample point, it is called a, Compound event., For example, in the experiment of “tossing a coin thrice” the events, E: ‘Exactly one head appeared’, F; ‘Atleast one head appeared’, G: ‘Atmost one head appeared’ etc., are all compound events. The subsets of S associated with these events are, E={HTT,THT,TTH}, F={HTT,THT, TTH, HHT, HTH, THH, HHH}, G= {TTT, THT, HTT, TTH}, Each of the above subsets contain more than one sample point, hence they are all, compound events., , 16.3.3 Algebra of events In the Chapter on Sets, we have studied about different, ways of combining two or more sets, viz, union, intersection, difference, complement, of a set etc. Like-wise we can combine two or more events by using the analogous set, notations., , Let A, B, C be events associated with an experiment whose sample space is S., , 1. Complementary Event For every event A, there corresponds another event, A‘ called the complementary event to A. It is also called the event ‘not A’., , For example, take the experiment ‘of tossing three coins’. An associated sample, space is, , S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}, , Let A={HTH, HHT, THH} be the event ‘only one tail appears’, , Clearly for the outcome HTT, the event A has not occurred. But we may say that, the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say, that ‘not A’ occurs., , 2021-22

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390 MATHEMATICS, , Thus the complementary event ‘not A’ to the event A is, A’ = {HHH, HTT, THT, TTH, TTT}, , or A’={@:@e Sand m¢A}=S-A., , 2. The Event ‘A or B’ Recall that union of two sets A and B denoted by AU B, contains all those elements which are either in A or in B or in both., , When the sets A and B are two events associated with a sample space, then, ‘A UB’ is the event ‘either A or B or both’. This event ‘A U B’ is also called “A or B’., Therefore Event ‘A or B’ = AUB, , = {@:@e Aor me B}, , 3. The Eyent ‘A and B’ We know that intersection of two sets A 7 B is the set of, those elements which are common to both A and B. i.e., which belong to both, ‘A and B’., If A and B are two events, then the set AM B denotes the event ‘A and B’., Thus, ANB={@:@e Aand@e B}, , For example, in the experiment of ‘throwing a die twice’ Let A be the event, “score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then, , A= {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6) }, so AMB= {(6,5), (6,6)}, Note that the set A> B = {(6,5), (6,6) } may represent the event ‘the score on the first, throw is six and the sum of the scores is atleast 11’., 4. The Event ‘A but not B’ We know that A-B is the set of all those elements, which are in A but not in B. Therefore, the set A-B may denote the event ‘A but not, B’.We know that, A-B=ANB’, , Example 6 Consider the experiment of rolling a die. Let A be the event “getting a, prime number’, B be the event “getting an odd number’. Write the sets representing, the events (i) Aor B (ii) A and B (iii) A but not B (iv) ‘not A’., Solution Here S={1, 2,3, 4,5, 6}, A= {2, 3, 5} and B= {1, 3, 5}, Obviously, , @ ‘AorB’=AUB={1, 2, 3,5}, , (ii) ‘Aand B’=AMB= {3,5}, , (iii) ‘A but not B’ = A-B = {2}, , (iv) ‘not A’=A’ = {1,4,6}, , 2021-22

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PROBABILITY 391, , 16.3.4 Mutually exclusive events In the experiment of rolling a die, a sample space is, S = {1, 2, 3, 4, 5, 6}. Consider events, A ‘an odd number appears’ and B ‘an even, number appears’, Clearly the event A excludes the event B and vice versa. In other words, there is, no outcome which ensures the occurrence of events A and B simultaneously. Here, A= {1,3,5} and B = {2, 4, 6}, , Clearly AM B= 9, i.e., A and B are disjoint sets., , In general, two events A and B are called mutually exclusive events if the, occurrence of any one of them excludes the occurrence of the other event, i.e., if they, can not occur simultaneously. In this case the sets A and B are disjoint., , Again in the experiment of rolling a die, consider the events A ‘an odd number, appears’ and event B ‘a number less than 4 appears’, , Obviously A= {1, 3,5} and B = {1, 2, 3}, Now 3 € A as well as 3 € B, Therefore, A and B are not mutually exclusive events., Remark Simple events of a sample space are always mutually exclusive., 16.3.5 Exhaustive events Consider the experiment of throwing a die. We have, S = {1, 2,3, 4,5, 6}. Let us define the following events, , A: ‘a number less than 4 appears’,, B: ‘a number greater than 2 but less than 5 appears’, and C: ‘a number greater than 4 appears’., , Then A = {1, 2, 3}, B = {3,4} and C = {5, 6}. We observe that, AUBUC={I, 2, 3} U {3,4} vu {5, 6} =S., , Such events A, B and C are called exhaustive events. In general, if E, E, ee E, aren, events of a sample space S and if, , E, VE, VE; v...UE, =VE;=S, , then E,, E,, ..... E, are called exhaustive events.In other words, events E,, E,, ..., E,, are said to be exhaustive if atleast one of them necessarily occurs whenever the, experiment is performed., , Further, if F, 7 E, = fori #j i.e., events E, and E, are pairwise disjoint and, , n, UE, =S, then events Ee ee E, are called mutually exclusive and exhaustive, i=l sa, , events., , 2021-22