Class 11 Mathematics Probability MCQs Set 12

Question. Two fair dice are tossed. Let X be the event that the first die shows an even number and Y be the event that the second die shows an odd number. The two events X and Y are
(a) mutually exclusive
(b) independent and mutually exclusive
(c) dependent
(d) independent
Answer: (d) independent

Question. A Six faced fair dice is thrown until 1 comes, then probability that 1 comes in even number of trials is
(a) 3/11
(b) 5/11
(c) 3/4
(d) 7/9
Answer: (b) 5/11

Question. A and B alternately throw with a pair of dice. Who ever gets a sum of 7 points 1st will win the game. If A starts the game, the probability of his winning is
(a) 4/11
(b) 5/11
(c) 6/11
(d) 3/11
Answer: (c) 6/11

Question. A pair of dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes before 7 is
(a) 1/5
(b) 2/5
(c) 3/5
(d) 4/5
Answer: (b) 2/5

Question. A man alternately tosses a coin and throws a die beginning with coin. The probability that he gets a head before he gets 5 or 6 on the die is
(a) 1/4
(b) 1/2
(c) 3/4
(d) 1/8
Answer: (c) 3/4

Question. A and B alternately cut a card each from a pack of cards with replacement and pack is shuffled after each cut. If A starts the game and the game is continued till one cuts a spade, the respective probabilities of A and B cutting a spade are
(a) 1/3, 2/3
(b) 3/4, 1/4
(c) 4/7, 3/7
(d) 3/7, 4/7
Answer: (c) 4/7, 3/7

Question. There are four machines and it is known that exactly two of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is
(a) 1/3
(b) 1/6
(c) 1/2
(d) 1/4
Answer: (a) 1/3

Question. A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the 4th time on the 7th draw is
(a) 5/64
(b) 27/32
(c) 5/32
(d) 1/2
Answer: (c) 5/32

Question. A biased coin with probability P, \( (0 < P < 1) \) of heads is tossed unitl a head appear for the first time. If the probability that the number of tosses required is even is \( \frac{2}{5} \) then P =
(a) 2/5
(b) 3/5
(c) 2/3
(d) 1/3
Answer: (d) 1/3

Question. A bag contains 3 white, 3 black and 2 red balls. One by one 3 balls are drawn without replacing them. For only the 3rd ball to be red the probability is
(a) 1/28
(b) 3/28
(c) 5/28
(d) 7/28
Answer: (c) 5/28

Question. An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with ball of the same colour. The process is repeated. The probability that the third ball drawn is black is
(a) 15/29
(b) 11/30
(c) 7/30
(d) 23/30
Answer: (d) 23/30

PROBLEMS ON TOTAL PROBABILITY

Question. There are 12 unbiased coins in a bag. Out of them 4 coins have head on both the sides. One coin is selected from the bag at random and tossed. The probability of getting a head is
(a) 1/2
(b) 1/3
(c) 2/3
(d) 1/4
Answer: (c) 2/3

Question. H is one of the 6 horses entered for a race and is to be ridden by one of the two jokeys A and B. It is 2 to 1 that A rides H in which case all the horses are likely to win. If B rides H, his chance is trebled. Then the odds against H winning is
(a) 4 to 13
(b) 13 to 4
(c) 13 to 5
(d) 13 to 7
Answer: (c) 13 to 5

Question. The probability that in a year of 22nd century chosen at random, there will be 53 Sundays is
(a) 3/28
(b) 2/28
(c) 7/28
(d) 5/28
Answer: (d) 5/28

Question. There are 2 bags one of which contains 3 black and 4 white balls, while the other contains 4 black and 3 white balls. A die is cast, if face 1 or 3 turns up a ball in taken from the 1st bag and if any other face turns up a ball is taken from the second bag. The probability of choosing a black ball is
(a) 5/21
(b) 10/21
(c) 11/21
(d) 6/21
Answer: (c) 11/21

Question. An urn contains 5 white and 7 black balls. A second urn contains 7 white and 8 black balls. One ball is transfered from the 1st urn to the 2nd urn without noticing its colour. A ball is now drawn at random from the 2nd urn. The probability that it is white is
(a) 8/92
(b) 89/192
(c) 9/92
(d) 98/192
Answer: (b) 89/192

Question. An urn contains 6 white and 4 black balls. A fair die whose faces are numbered from 1 to 6 is rolled and number of balls equal to that of the number appearing on the die is drawn from the urn at random. The probability that all those are white is
(a) 1/5
(b) 2/5
(c) 3/5
(d) 4/5
Answer: (a) 1/5

Question. Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 black balls. One ball is drawn at random from A and placed in B. Then one ball is drawn at random from B and placed in A. If one ball is now drawn from A then the probability that it is found to be red is
(a) 32/55
(b) 33/55
(c) 32/63
(d) 25/66
Answer: (a) 32/55

Question. \( \frac{2}{3} \) of students of a class are boys and the rest girls. It is given that the probability of a girl getting 1st class is 0.25 and the same for a boy is 0.28. From that class a student is selected at random. The probability that the student is a 1st class student
(a) 0.25
(b) 0.26
(c) 0.27
(d) 0.28
Answer: (c) 0.27

Question. A bag contains (2n + 1) coins. It is known that n of these have a head on both the sides, whereas the remaining (n + 1) coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is 31/42, then value of n is [EAMCET 2013]
(a) 10
(b) 8
(c) 6
(d) 25
Answer: (a) 10

PROBLEMS ON BAYE’S THEOREM

Question. For k = 1,2,3 the box \( B_k \) contains k red balls and (k+1) white balls. Let \( P(B_1) = \frac{1}{2} \), \( P(B_2) = \frac{1}{3} \), \( P(B_3) = \frac{1}{6} \). A box is selected at random and a ball is drawn from it. If a red ball is drawn then the probability that it has come from box \( B_2 \) is [EAMCET 2008]
(a) 35/78
(b) 14/39
(c) 10/13
(d) 12/13
Answer: (b) 14/39

Question. A letter is known to have come from TATANAGAR or CALCUTTA. On the envelope just two consecutive letters TA are visible. The probability that the letter has come from CALCUTTA is
(a) 4/11
(b) 7/11
(c) 1/22
(d) 21/22
Answer: (a) 4/11

Question. A man is known to speak the truth 2 out of 3 times. He throws a die and reports that it is ‘Five’. The probability that it is actually not a ‘Five’ is
(a) 1/2
(b) 2/7
(c) 1/7
(d) 5/7
Answer: (d) 5/7

Question. Out of ten coins, one of the coin is known to have heads on both sides. He takes out one coin at random and tosses it 5 times. If it always falls with head upwards, the probability that it is double-headed coin is
(a) 32/51
(b) 32/41
(c) 32/61
(d) 12/41
Answer: (b) 32/41

ADDITIONAL PROBLEMS

Question. If 4 people are chosen at random, then the probability that no two of them were born on the same day of the week is
(a) \( \frac{^7P_4}{7^4} \)
(b) \( \frac{^7C_4}{7^4} \)
(c) \( \frac{7!}{7^4} \)
(d) \( \frac{1}{7^4} \)
Answer: (a) \( \frac{^7P_4}{7^4} \)

Question. What is the probability that the birthdays of six people will fall in exactly two calender months
(a) \( \frac{^{12}C_2 \cdot (2^6 - 2)}{(12)^6} \)
(b) \( \frac{^{12}C_2 \cdot (2^6 + 2)}{(12)^6} \)
(c) \( \frac{^{12}C_2 \cdot (2^6 - 4)}{(12)^6} \)
(d) \( \frac{^{12}C_2 \cdot (2^6)}{(12)^6} \)
Answer: (a) \( \frac{^{12}C_2 \cdot (2^6 - 2)}{(12)^6} \)

Question. The odds that book be reviewed favourably by three independent critics are 5 to 2, 4 to 3 and 3 to 4 respectively. The probability that of the three reviews a majority will be favourable is
(a) 209/343
(b) 135/343
(c) 60/343
(d) 120/343
Answer: (a) 209/343

Question. Set S has 4 elements, A and B are subsets of S. The probability that A and B are not disjoint is
(a) 175/256
(b) 173/256
(c) 85/128
(d) 45/64
Answer: (a) 175/256

Question. The sum of two natural numbers is 20. Find the chance that their product less than 50 is
(a) 4/19
(b) 3/19
(c) 2/19
(d) 1/19
Answer: (a) 4/19

Question. In constructing problem on vectors, the three components of a vector are randomly chosen from the digit 0 to 5 with replacement. The probability that the magnitude of the vector is 5 is
(a) 1/24
(b) 1/12
(c) 1/6
(d) 1/30
Answer: (a) 1/24

Question. Three newly wedded couples are dancing at a function. If the partner is selected at random the chance that at least one husband is not dancing with his own wife is
(a) 1/3
(b) 1/6
(c) 5/6
(d) 2/3
Answer: (c) 5/6

Question. Team A plays with 5 other teams exactly once. Assuming that for each match the probabilities of a win, draw and loss are equal, then
(a) the probability that A wins and loses equal number of matches is 34/81
(b) the probability that A wins and loses equal number of matches is 17/81
(c) the probability that A wins more number of matches than it loses is 17/81
(d) the probability that A loses more number of matches than it wins is 16/81
Answer: (b) the probability that A wins and loses equal number of matches is 17/81

Question. Two fair dice are rolled simultaneously. It is found that one of them shows odd prime number. The probability that remaining die also shows an odd prime number, is equal to
(a) 1/5
(b) 2/5
(c) 3/5
(d) 4/5
Answer: (a) 1/5

Question. Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up respectively is
(a) 16/31
(b) 1/2
(c) 17/31
(d) 18/31
Answer: (a) 16/31

Question. E, \( \overline{E} \) are mutually exclusive and exhaustive events in a sample space 'S' then maximum value of P(E). P(\( \overline{E} \)) is
(a) 1/2
(b) 1/3
(c) 1/4
(d) 1/5
Answer: (c) 1/4

Question. In a department there are two Professors, four Readers and 6 Lecturers. A committee of three persons is to be formed out of the staff of the department. The probability that the committee consists of at least two lecturers is
(a) 1/4
(b) 1/3
(c) 1/2
(d) 1/8
Answer: (c) 1/2

Question. 10 gentlemen and 6 ladies are to sit for a dinner at a round table. The probability that no two ladies sit together is
(a) \( \frac{9! \times 5!}{15!} \)
(b) \( \frac{9! \times {}^{10}P_6}{15!} \)
(c) \( \frac{9! \times {}^{10}C_6}{15!} \)
(d) \( \frac{9! \times {}^{10}C_5}{15!} \)
Answer: (b) \( \frac{9! \times {}^{10}P_6}{15!} \)

Question. 100 tickets are numbered as 00, 01, 02,... 09, 10, 11, 12, .....,99 out of them one ticket is drawn at random. The probability that the sum of the digits of the number on the ticket is 9 is
(a) 7/100
(b) 9/100
(c) 1/10
(d) 1/100
Answer: (c) 1/10

Question. If 5 biscuits are distributed among 3 beggers, the chance that a particular beggar will get 2 biscuits is
(a) 80/243
(b) 30/125
(c) 2/15
(d) 3/10
Answer: (a) 80/243

Question. From a box containing 10 cards numbered 1 to 10, four cards are drawn together. The probability that their sum is odd is
(a) 1/2
(b) 11/21
(c) 10/21
(d) 1/21
Answer: (c) 10/21