{"id":16910,"date":"2020-01-02T15:13:43","date_gmt":"2020-01-02T09:43:43","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=16910"},"modified":"2021-07-02T15:32:57","modified_gmt":"2021-07-02T10:02:57","slug":"1st-puc-maths-question-bank-chapter-16","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank-chapter-16\/","title":{"rendered":"1st PUC Maths Question Bank Chapter 16 Probability"},"content":{"rendered":"

Students can Download Maths Chapter 16 Probability Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 1st PUC Maths Question Bank Chapter 16 Probability<\/h2>\n

Question 1.
\nName the mathematician who developed the theory of probability in 1933.
\nAnswer:
\nTheory of probability was developed by Russian mathematician A.N. Kolmogorv in 1933.<\/p>\n

Question 2.
\nDefine the following terms:
\n(i) Random experiment
\n(ii) Outcomes
\n(iii) Sample space
\n(iv) Sample point.
\nAnswer:
\n(i) Random experiment: An experiment is called random experiment if it has more than one possible outcome and it is not possible to predicts the outcome in advance.\u00a0\u00a0\u00a0\u00a0 .<\/p>\n

(ii) Outcomes: A possible result of a random experiment is called its outcome.<\/p>\n

(iii) Sample space: The set of all possible outcomes of a random experiment is called the sample space associated with the experiment. It is denoted by the symbol S.<\/p>\n

(iv) Sample point: Each element of a sample space is called sample point, or each outcome of the random experiment is called sample point.<\/p>\n

\"KSEEB<\/p>\n

Question 3.
\nDescribe the sample space for the indicated experiment, (i) A coin is tossed three times.
\nAnswer:
\nSample space = S
\n= {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}<\/p>\n

(ii) A die is thrown two times.
\nAnswer:
\nRequired sample space = S
\n= {(x,y) : x,y \u2208 {1,2,3,4,5,6}}
\n= {(x,y) : x,y = 1,2,3,4,5,6}
\n= {(1.1). (1,2)(1,3), (1,4), (1,5), (1,6),
\n= (2,6), ……………… , (3,1), …………….. , (3,6), (4,1), ……………. , (4,6),
\n(5,6), …………. (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}<\/p>\n

(iii) A coin is tossed four times.
\nAnswer:
\nS = {HHHH, HHHT, HHTH, HTHH,THHH, HHTT, HTHT, HTTH, THHT, THTH,
\nTTHH, HTTT, THTT,TTHT, TTTH ,TTTT}<\/p>\n

(iv) A coin is tossed and then a die is rolled only in case a head is shown on the coin.
\nAnswer:
\nRequired sample space = S = {H1,H2,H3,H4,H5,H6,T}<\/p>\n

(v) A coin is tossed and a die is thrown.
\nAnswer:
\nRequired sample space = S
\n= {(x, y): x \u2208 {H, T}, y \u2208 {1,2,3,4,5,6}}
\n= {(H, 1), (H ,2),(H ,3), (H ,4) (H ,5), (H ,6),
\n(T, 1)(T, 2)(T, 3), (T, 4), (T, 5), (T, 6)}
\n= {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}<\/p>\n

Question 4.
\n2 boys and 2 girls are in room X and 1 boy and 3 girls are in room Y. Specify the sample space for the experiment in which a room is selected and then a person.
\nAnswer:
\nLet boys be denoted by BX<\/sub>, B2<\/sub>, B3<\/sub> and girls be G1<\/sub>, G2<\/sub>, G3<\/sub>, GA<\/sub>, G5<\/sub>
\nThen the required sample space = S.
\n= {XB1<\/sub>, XB2<\/sub>, XG, XG2<\/sub>, YB, YG, YGA<\/sub>, YGS<\/sub>}
\nHere B1<\/sub>, B1<\/sub>, G1<\/sub>, G2<\/sub> are associated with X and B3<\/sub>, Gb<\/sub>, Ga<\/sub>, Gs<\/sub> are associated with Y.<\/p>\n

\"KSEEB<\/p>\n

Question 5.
\nOne die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.
\nAnswer:
\nLet R,W,B be the red die, white die, blue die respectively.
\n\u2234 Required sample space = S
\n= {R1, R2, R3, R4, R5, R6,
\nW1, W2, W3, W4, W5, W6,
\nB1, B2, B3, B4, B5, B6}<\/p>\n

Question 6.
\nAn experiment consists of recording boy – girl composition of family with 2 children,
\n(i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?
\n(ii) What is the sample space if we are interested in the number of girls in the family?
\nAnswer:
\nLet B stands for boys and G stands for girl. Then
\n(i) Required sample space = S
\n= {BB, BG, GB, GG}
\nRequired sample space = {0, 1, 2}<\/p>\n

Question 7.
\nA box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.
\nAnswer:
\nLet R stands for red ball and W stands for white ball. Then required sample space = S
\n= {RW, WR, WW}
\n= {R,W),(W,R),(W,W)}<\/p>\n

Question 8.
\nAn experiment consists of tossing a coin and then throwing it second time if a head occurs. If tail occurs on the first toss, then a die is rolled once. Find the sample space.
\nAnswer:
\nSample space = S
\n= {HH, HT, T1, T2,T3, T4, T5, T6}<\/p>\n

\"KSEEB<\/p>\n

Question 9.
\nSuppose 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.
\nAnswer:
\nRequired sample space = S
\n= {DDD, DDN, DND, NDD, NNN, NND,NDN,DNN}<\/p>\n

Question 10.
\nA coin is tossed. If the outcome 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?
\nAnswer:
\nRequired sample space = S
\n= {T, H1, H3, H5, H21, H22, H23, H24, H25, H26,
\nH41, H42, H43, H44, H45, H46,
\nH61, H62, H63, H64, H65, H66}<\/p>\n

Question 11.
\nThe numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are put in the box and mixed thoroughly. A person draws two slips from the box one after the other without replacement. Write the sample space for the experiment.
\nAnswer:
\nRequired sample space – S
\n= {(1,2), (1,3), (1,4), (2,1), (2,3),
\n(2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)}<\/p>\n

Question 12.
\nAn experiment 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.
\nAnswer:
\nRequired sample space = 5
\n= {2H, 2T, 4H, 6H, 6T, 1HH
\n1HT, 1TH, 1TT, 3HH, 3HT,
\n3TH, 3TT, 5HH, 5HT 5TH, 5TT}<\/p>\n

\"KSEEB<\/p>\n

Question 13.
\nA coin is tossed. If it shows 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?
\nAnswer:
\nRequired sample space = S = {TR1, TR2, TB1, TB2, TB3, H1, H2, H3, H4, H5, H6}
\nHere R1<\/sub>,R2<\/sub> are two red balls and B1<\/sub>,B2<\/sub>,B3<\/sub> are three black balls.<\/p>\n

Question 14.
\nDefine the following terms,
\n(i) Event
\n(ii) Impossible event
\n(iii) Sure event
\n(iv) Simple event (Elementary event)
\n(v) Compound event.
\nAnswer:
\n(i) Event: Any subset E of a sample space S is called an event.
\n(ii) Impossible even: The empty set \u03c6 describes an events. Then \u03c6 is called an impossible event.
\n(iii) Sure event: The sample space S describes an event. Then S, sample space is called the sure event.
\n(iv) Simple event: If an event E has only one sample point of a sample space, it is called a simple event.
\n(v) Compound event: If an event has more than one sample point, it is called a compound event.<\/p>\n

\"KSEEB<\/p>\n

Question 15.
\nDefine: Complementary event, the event \u2018A or B\u2019, the event \u2018A and B\u2019, the event \u2018A but not B\u2019, mutually exclusive events, exhaustive events and mutually exclusive and exhaustive events.
\nAnswer:
\n(i) Complementary event (\u2018not event\u2019): Let A be event associated with an experiment whose sample space is S. Then, the complementary event \u2018not A\u2019 to the event A is denoted by A’ and is defined as A’ = { \u03c9 : \u03c9 \u2208 S and \u03c9 \u2209 A} = S-A<\/p>\n

(ii) The even \u2018A or B\u2019: The sets A and B are two events associated with a sample space, then \u2019A \u222a B\u2019 is the event \u2018either A or B or both\u2019, this event is called\u2018A or B\u2019.
\ni.e., \u2019A or B’ = A \u222a B = {\u03c9 : (\u03c9 \u2208 A or \u03c9 \u2208 B]<\/p>\n

(iii) The event \u2018A and B\u2019: If A and B are two events, then the set A \u2229 B denotes the event \u2018A and B\u2019.
\n\u2234 \u2019A and B\u2019= A \u2229 B = {\u03c9 : \u03c9 \u2208 A and \u03c9 \u2208 B}<\/p>\n

(iv) The event \u2018A but not B\u2019: If A and B are two events associated with an experiment, then the set A – B denotes the event \u2018A but not B\u2019.
\nNote: A-B = A \u2229 B’<\/p>\n

(v) Mutually exclusive events: If A and B are two events associated with an experiment, then A and B are called mutually exclusive events if they cannot occur simultaneously. In this case A \u2229 B = \u03d5<\/p>\n

Note: A and B are disjoint sets.
\n\"1st<\/p>\n

Question 16.
\nConsider the experiment of rolling a die. Let A be the event \u2018getting a prime number, B be the event \u2018getting an odd number\u2019. Write the sets representing the events”
\n(i) \u2018A or B\u2019
\n(ii) \u2018A and B\u2019
\n(iii) \u2018not A\u2019
\n(iv) \u2018A but not B\u2019.
\nAnswer:
\nWe have, the sample space of an experiment of rolling a die is S = {1,2,3,4,5,6}
\nHere, A = {2,3,5} and B = {1,3,5}
\n(i) \u2019A or B\u2019=A \u222a B = {1,2,3,5}
\n(ii) \u2019A and B\u2019=A \u2229 B = {3,5}
\n(iii) \u2019not A\u2019= A’ = S – A = {1,4,6}
\n(iv) \u2019A but not B \u2019= A – B = {2}<\/p>\n

\"KSEEB<\/p>\n

Question 17.
\nTwo dice are thrown and the sum of the numbers which comes up on the dice is noted. Let us consider the following events associated with this experiment.
\nA : \u2018the sum is even\u2019
\nB : ‘the sum is a multiple of 3’
\nC: \u2018the sum is less than 4\u2019
\nD: \u2018the sum is greater than 11\u2019.
\nWhich pairs of these events are mutually exclusive?
\nAnswer:
\nWe have, sample space
\n= S = {(x,y) : x,y =1, 2, 3, 4, 5, 6}
\ni.e., 5 = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
\n(2,1),………… (2,6), (3,1), ……….. ,(3,6),
\n(4,6),……….. (5,1), …………… (5,6),
\n(6,1),(6,2),(6,3),(6,4), (6,5),(6,6)}<\/p>\n

Now, A = {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6)
\n(3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,4), (6,6)}<\/p>\n

B = {(1,2), (1,5), (2,4), (3,3), (3,6), (4,5),
\n(2,1),(4,2),(5,4),(6,3),(6,6)}<\/p>\n

C = {(1,1),(1,2),(2,1)}<\/p>\n

D = {(6,6)}
\nWe find that,
\nA \u2229 E = {(1,5), (2,4), (3,3), (4,2), (5,1), (6,6)}
\n\u21d2 A \u2229 B \u2260 \u03d5
\n\u21d2 A and B are not mutually exclusive events.
\nSimilarly,
\nA \u2229 C \u2260 \u03d5, A \u2229 D \u2260 \u03d5
\nB \u2229 C\u2260 \u03d5,B \u2229 D \u2260 \u03d5, C \u2229 D\u2260 \u03d5
\nThus the pairs (A, C), (A, D), (B, C) and (B, D) are not mutually exclusive events and pair (C, D) are mutually exclusive events.<\/p>\n

Question 18.
\nA coin is tossed three times, consider the following events,
\nA : \u2018no head appears\u2019.
\nB: \u2018exactly one head appears\u2019
\nC: \u2018at least two heads appear\u2019
\nDo they form a set of mutually exclusive and exhaustive events?
\nAnswer:
\nThe sample space of the experiment is,
\nS ={HHH, HHT, HTH, THH, ITT, TTH, THT, HTT}
\nHere, A = {m},B = {TTH, THT, HTT}
\nC = {HHT, HTH, THH, HHH}
\nNow, A \u222a B \u222a C = {TTT, TTH, THT, HTT, HHT, HTH, THH, HHH}
\n\u21d2 A \u222a B \u222a C = S
\n\u21d2 A, B and C are exhaustive events.
\nAlso A\u2229B = \u03d5,A\u2229C = \u03d5 and B\u2229C = \u03d5
\n\u21d2 A, B and C are mutually exclusive.
\nHence A, B and C are mutually exclusive and exhaustive events.<\/p>\n

\"KSEEB<\/p>\n

Question 19.
\nA die is rolled. Let E be the event \u201cdie shows 4\u201d and F be the event \u201cdie shows even number\u201d. Are E and F mutually exclusive?
\nAnswer:
\nThe sample space of an experiment is,
\n5 = {1,2,3,4,5,6}
\nHere, E = {4} and F = {2,4,6}
\nNow, E \u2229F = {4} \u2260 \u03d5
\n\u21d2 E and F are not mutually exclusive.<\/p>\n

Question 20.
\nA die is thrown. Describe the following events;
\n(i) A: a number less than 7.
\n(ii) B: a number greater than 7
\n(iii) C: a multiple of 3.
\n(iv) D: a number less than 4.
\n(v) E : an even number greater than 4.
\n(vi) F: a number not less than 3. Also find A \u222a B, A \u2229 B, B \u222a C, E \u2229 F, D \u2229 E, A – C, D – E,
\nE \u2229 F\u2019
\nAnswer:
\n\"1st<\/p>\n

Question 21.
\nAn experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater than 8, B: 2 occurs on either die. C: the sum is atleast 7 and a multiple of 3. Which pairs of these events are mutually exclusive?
\nAnswer:
\nWe have, sample space
\nS ={(x,y) : x,y = 1,2,3,4,5,6}
\nThen, A = {(3,6), (4,5), (4,6), (5,4), (5,5),
\n(5,6), (6,3), (6,4), (6,5), (6,6)}<\/p>\n

B = {(1,2), (2,2), (3,2), (4,2), (5,2),
\n(6,2) , (2,1), (2,3), (2,4), (2,5), (2,6)}<\/p>\n

C = {(3,6), (4,5), (5,4), (6,3), (6,6)}
\nNow A \u2229 B = { } = \u03d5
\nA \u2229 C = {(3,6), (4,5), (5,4), (6,3), (6,6)}
\nB \u2229 C = { } = \u03d5
\n\u2234 A and B, B and C are mutually exclusive events.<\/p>\n

\"KSEEB<\/p>\n

Question 22.
\nThree coins are tossed once. Let A denote the event \u2018three heads show\u2019, B denote the event \u2018two heads and one tail show\u2019, C denote the event \u2018three tails show\u2019 and D denote the event \u2018a head shows on the first can\u2019. Which events are
\n(i) mutually exclusive?
\n(ii) simple ?
\n(iii) compound?
\nAnswer:
\nWe have, sample space
\n= S = {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT)
\nThen
\nA = {HHH}
\nB = {HHT, HTH, THH}
\nC = {T,T,T}
\nD = {HHH, HTH, HHT, HTT}<\/p>\n

(i) Clearly A \u2229 B = \u03d5, A \u2229 C = \u03d5, C \u2229 D = \u03d5
\n\u21d2 A and B ,A and C ; C and D are mutually exclusive.
\n(ii) A, C are simple events.
\n(iii) B, D are compounds events.<\/p>\n

Question 23.
\nThree coins are tossed. Describe (i) Two events which are mutually exclusive, (ii) Three events which are mutually exclusive and exhaustive, (iii) Two events, which are not ‘ mutually exclusive (iv) Two events which are mutually exclusive but not exhaustive, (v) Three events which are mutually exclusive but not exhaustive.
\nAnswer:
\nLet (i)
\nA: \u2018atleast two heads appear\u2019
\nB: \u2018atleast two tails appear\u2019
\nHere, A = {HHT, HTH, THH, HHH}
\nB = {TTH, THT, HTT, TTT}
\n\u2234 A \u2229 B = \u03d5 , \u21d2 A, B mutually exclusive.
\n(Note: Give any suitable example (events) associated with experiment)<\/p>\n

(ii) Let A : \u2018No tail appears\u2019
\nB: \u2018Exactly one tail appears\u2019
\nC: \u2018Atleast two tails appear\u2019
\nHere A = {HHH]
\nB = {THH, HTH, HHT}
\nC = {TTH, THT, HTT, TTT}
\nClearly, A \u222a B \u222a C = sample space = S and A \u2229 B = \u03d5 , B \u2229 C = \u03d5, A \u2229 C
\n\u2234 A, B, C are mutually exclusive and mutually exclusive and exhaustive events.<\/p>\n

\"KSEEB<\/p>\n

(iii) Let
\nA: \u2018almost two tails appear\u2019
\nB: \u2018exactly two tails appear\u2019
\nHere,
\nA = {HHH, THH, HTH, HHT, HTT, THT, TTH}
\nB = {HTT,THT,TTH}
\nClearly, A \u2229 B \u2260 \u03d5
\n\u2234 A and B are not mutually exclusive
\n(iv) \u2018Getting exactly one head\u2019 and \u2018Getting exactly two heads\u2019. (Try the explanation)
\n(v) \u2018Getting exactly one tail\u2019, \u2018Getting exactly two tails\u2019 and \u2018Getting exactly three tails\u2019. (Try explanation)<\/p>\n

Question 24.
\nTwo dice are thrown. The events A, B, and C are as follows. A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice < 5. Describe the events: (i) A’ (ii) not B (iii) A or B (iv) A and B (v) A but not C (vi) B or C (vii) B and C (viii) A\u2229B’\u2229C’.
\nAnswer:
\nWe have, sample space S of the experiment is 5={(x,y):x,y = l,2,3,4,5,6}
\nThen A = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
\n(4.1), (4,2), (4,3), (4,4), (4,5), (4,6)
\n(6.1),(6,2),(6,3),(6,4),(6,5),(6,6)}<\/p>\n

B = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
\n(3.1) , (3,2), (3,3), (3,4), (3,5), (3,6),
\n(5.1) , (5,2), (5,3), (5,4), (5,5), (5,6)}<\/p>\n

C = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3)
\n(3.1) , (3,2), (4,1)}<\/p>\n

(i) A’ = B (ii) B\u2019 = A (iii) A\u222aB = S (iv) A\u2229B = \u03d5 (v) A – C = {(2,4),(2,5),(2,6),(4,2),(4,3) , (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4),(6,5),(6,6)}<\/p>\n

(vi) B \u222a C = {(1,1), (1,2), ………… (1,6), (2,1), (2,2),
\n(2.3),(3,1), (3,2), (3,3), ……….. (3,6), (4,1), (5,1), (5,2),(5,6)}<\/p>\n

(vii) B \u2229 C = {(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}<\/p>\n

\"KSEEB<\/p>\n

(viii) Here, C’ = S – C and B’ = A
\n\u2234 A \u2229 B’\u2229 C’ = A \u2229 A \u2229 C’
\n= A \u2229 C’
\n= {(2,4), (2,5), (2,6), (4,2),
\n(4.3), (4,4), …………, (4,6), (6,1), (6,2), ………. , (6,6)}<\/p>\n

Axiomatic Approach to Probability<\/strong><\/p>\n

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 and range is the interval [0, 1]. Satisfy the following axioms.<\/p>\n