Class 11 Mathematics Probability MCQs Set 09

Question. A number is chosen from the first 100 natural numbers. The probability that it is a multiple of 4 or 6 is
(a) \( \frac{25}{100} \)
(b) \( \frac{16}{100} \)
(c) \( \frac{8}{100} \)
(d) \( \frac{33}{100} \)
Answer: (d) \( \frac{33}{100} \)

Question. The results of an examination in two papers A and B for 20 candidates were recorded as follows. 8 passed in paper A, 7 passed in paper B, 8 failed in both the papers A and B. If one is selected at random, the probability that the candidate has failed in A or B is
(a) \( \frac{15}{20} \)
(b) \( \frac{16}{20} \)
(c) \( \frac{17}{20} \)
(d) \( \frac{18}{20} \)
Answer: (c) \( \frac{17}{20} \)

Question. In a class there are 10 men & 20 women. Out of them half of the number of men & half of the number of women have brown eyes. Out of them if a person is chosen at random, the chance that for the person chosen to be a man or brown eyed person is
(a) \( \frac{1}{3} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{3}{4} \)
(d) \( \frac{1}{4} \)
Answer: (b) \( \frac{2}{3} \)

Question. If the probabilities of two dogs A and B dying within 10 years are respectively p and q, then the probability that at least one of them will be alive at the end of 10 years is
(a) \( p + q \)
(b) \( 1 - pq \)
(c) \( p + q - pq \)
(d) \( pq \)
Answer: (b) \( 1 - pq \)

Question. The probability that a man A will be alive for 20 more years is \( \frac{3}{5} \) and the probability that his wife will be alive for 20 more years is \( \frac{2}{3} \). The probability that only one will be alive at the end of 20 years is
(a) \( \frac{4}{15} \)
(b) \( \frac{2}{15} \)
(c) \( \frac{7}{15} \)
(d) \( \frac{8}{15} \)
Answer: (c) \( \frac{7}{15} \)

Question. A husband and wife appear in an interview for two vacancies in the same post. The probability of husbands selection is \( \frac{1}{7} \) and that of wife is \( \frac{1}{5} \). The probability that both of them will be selected is
(a) \( \frac{24}{35} \)
(b) \( \frac{2}{7} \)
(c) \( \frac{1}{35} \)
(d) \( \frac{2}{35} \)
Answer: (c) \( \frac{1}{35} \)

Question. A can hit a target 3 times in 6 shots; B, 2 times in 4 shots and C, 4 times in 4 shots. All of them fire at a target independently. The probability that the target will be hit is
(a) 1
(b) 0.5
(c) 0.25
(d) 0.125
Answer: (a) 1

PROBLEMS ON CONDITIONAL PROBABILITY

Question. If A and B are two events such that \( P(A) = \frac{3}{8} \), \( P(B) = \frac{5}{8} \) and \( P(A \cup B) = \frac{3}{4} \), then \( P\left(\frac{B}{A}\right) = \)
(a) \( \frac{2}{5} \)
(b) \( \frac{3}{5} \)
(c) \( \frac{4}{5} \)
(d) \( \frac{1}{5} \)
Answer: (b) \( \frac{3}{5} \)

Question. If A and B are two independent events such that \( P(A) = \frac{1}{3} \) and \( P(B) = \frac{3}{4} \), then \( P\left\{\frac{B}{(A \cup B)}\right\} = \)
(a) \( \frac{7}{10} \)
(b) \( \frac{8}{10} \)
(c) \( \frac{9}{10} \)
(d) \( \frac{6}{10} \)
Answer: (c) \( \frac{9}{10} \)

Question. \( E_{1}, E_{2} \) are events of a sample space such that \( P(E_{1}) = \frac{1}{4}, P\left(\frac{E_{2}}{E_{1}}\right) = \frac{1}{2}, P\left(\frac{E_{1}}{E_{2}}\right) = \frac{1}{4} \). Then \( P\left(\frac{\overline{E_{1}}}{\overline{E_{2}}}\right) = \)
(a) \( \frac{1}{3} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{4} \)
Answer: (d) \( \frac{3}{4} \)

Question. A six - faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be 7. The probability that the number 3 has appeared at least once is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{5} \)
Answer: (b) \( \frac{1}{3} \)

Question. The results of students of a college revealed the following facts. 25% of students failed in Mathematics, 15% of students failed in Chemistry, 10% of students failed in both. If a student is selected at random. The probability that he has failed in Mathematics, given that he failed in Chemistry is
(a) \( \frac{1}{5} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{3}{5} \)
(d) \( \frac{1}{3} \)
Answer: (b) \( \frac{2}{3} \)

Question. Suppose E and F are two events of a random experiment. If the probability of occurrence of E is 1/5 and the probability of occurence of F given E is 1/10, then the probability of non-occurrence of at least one of the events E and F is
(a) 1/18
(b) 1/2
(c) 49/50
(d) 1/50
Answer: (c) 49/50

Question. The probability of drawing 2 red balls in succession from a bag containing 4 red balls and 5 black balls when the ball that is drawn first is not replaced is
(a) \( \frac{1}{6} \)
(b) \( \frac{16}{81} \)
(c) \( \frac{3}{8} \)
(d) \( \frac{4}{9} \)
Answer: (a) \( \frac{1}{6} \)

Question. A box contains 5 black, 4 white and 6 red balls. Two balls are drawn without replacement. The probability that the first will be white and the second will be black is
(a) \( \frac{1}{21} \)
(b) \( \frac{2}{21} \)
(c) \( \frac{3}{21} \)
(d) \( \frac{4}{21} \)
Answer: (b) \( \frac{2}{21} \)

PROBLEMS ON INDEPENDENT EVENTS

Question. If A and B are two events such that \( P(A \cup B) = \frac{5}{6} \), \( P(A \cap B) = \frac{1}{3}, P(A) = \frac{2}{3} \), then A and B are
(a) dependent events
(b) independent events
(c) mutually exclusive events
(d) mutually exclusive and independent events
Answer: (b) independent events

Question. If A and B are two independent events such that \( P(B) = \frac{2}{7} \), \( P(A \cup B^{c}) = 0.8 \) then P(A)=
(a) 0.1
(b) 0.2
(c) 0.3
(d) 0.4
Answer: (c) 0.3

Question. If A and B are two independent events, and P(A) = 1/4; P(B) = 1/3 then, P((A – B) \( \cup \) (B – A)) =
(a) \( \frac{5}{12} \)
(b) \( \frac{7}{12} \)
(c) \( \frac{11}{12} \)
(d) \( \frac{1}{3} \)
Answer: (a) \( \frac{5}{12} \)

Question. A and B are two candidates seeking admission in I.I.T. The probability that both A and B are selected is at most 0.4. If the probability of A's selection is 0.5, then the probability of B's selection if A and B are independent is
(a) 0.6
(b) <0.6
(c) \( \leq 0.6 \)
(d) >0.6
Answer: (c) \( \leq 0.6 \)

Question. A speaks truth in 75% of the cases and B in 80% of the cases. The percentage of cases they are likely to contradict each other in making the same statement is
(a) 25%
(b) 35%
(c) 50%
(d) 65%
Answer: (b) 35%

Question. 7 coupons are numbered 1 to 7. Four are drawn one by one with replacement. The probability that the least number appearing on any selected coupon is greater than or equal to 5 is
(a) \( \left(\frac{3}{7}\right)^{4} \)
(b) \( \frac{6}{7^{3}} \)
(c) \( \frac{4}{7^{3}} \)
(d) \( \left(\frac{3}{4}\right)^{4} \)
Answer: (a) \( \left(\frac{3}{7}\right)^{4} \)

Question. A box contains 10 mangoes out of which 4 are rotten. Two mangoes are taken together. If one of them is found to be good, the probability that the other is also good is
(a) \( \frac{1}{3} \)
(b) \( \frac{8}{13} \)
(c) \( \frac{5}{13} \)
(d) \( \frac{2}{3} \)
Answer: (c) \( \frac{5}{13} \)

Question. Counters numbered 1, 2, 3 are placed in a bag and one is drawn at random and replaced. The operation is being repeated three times. The probability of obtaining a total of 6 is
(a) \( \frac{7}{9} \)
(b) \( \frac{6}{27} \)
(c) \( \frac{7}{27} \)
(d) \( \frac{5}{27} \)
Answer: (c) \( \frac{7}{27} \)

Question. A fair die is tossed twice. The probability of getting a 4, 5 or 6 on the 1st toss and a 1, 2, 3 or 4 on the 2nd toss is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{8} \)
Answer: (b) \( \frac{1}{3} \)

Question. An ordinary die has 4 blank faces, one face marked 2 and an other marker 3 Then the probability of obtaining a total of exactly 5 in 2 throws is
(a) \( \frac{1}{36} \)
(b) \( \frac{1}{18} \)
(c) \( \frac{1}{9} \)
(d) \( \frac{1}{72} \)
Answer: (b) \( \frac{1}{18} \)

Question. A card is drawn from a well shuffled pack of 52 playing cards. It is replaced in the pack after noting its colour. Again a card is drawn at random. The probability that the 1st card drawn may be a heart and the second card drawn may not be a queen is
(a) \( \frac{5}{13} \)
(b) \( \frac{4}{13} \)
(c) \( \frac{3}{13} \)
(d) \( \frac{2}{13} \)
Answer: (c) \( \frac{3}{13} \)

Question. A pack of cards is distributed to four players as in the game of bridge. The probability that a particular player will not get an ace in three consecutive games is
(a) \( \frac{^{48}c_{13}}{^{52}c_{13}} \times 3 \)
(b) \( \left(\frac{^{48}c_{13}}{^{52}c_{13}}\right)^{3} \)
(c) \( \frac{^{48}c_{13}}{^{52}c_{13}} \)
(d) \( \left(\frac{3 \times ^{48}c_{13}}{^{52}c_{13}}\right)^{3} \)
Answer: (b) \( \left(\frac{^{48}c_{13}}{^{52}c_{13}}\right)^{3} \)

Question. The probability of drawing 4 white and 2 black balls in two drawings in succession from a bag containing 1 red, 4 black and 6 white balls, if the drawing is without replacement is
(a) \( \frac{^{6}c_{4}}{^{11}c_{4}} \times \frac{^{4}c_{2}}{^{11}c_{2}} \)
(b) \( \frac{^{6}c_{4}}{^{11}c_{4}} \times \frac{^{4}c_{2}}{^{7}c_{2}} \)
(c) \( \frac{^{10}c_{6}}{^{11}c_{6}} \)
(d) \( \frac{^{10}c_{2}}{^{11}c_{2}} \)
Answer: (b) \( \frac{^{6}c_{4}}{^{11}c_{4}} \times \frac{^{4}c_{2}}{^{7}c_{2}} \)

Question. The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. If both of them try independently, the probability that the problem will be solved is
(a) \( \frac{11}{21} \)
(b) \( \frac{13}{21} \)
(c) \( \frac{16}{21} \)
(d) \( \frac{8}{21} \)
Answer: (c) \( \frac{16}{21} \)

ADDITIONAL PROBLEMS

Question. A committee of 6 persons is to be formed from a group of 7 gents and 4 ladies. What is the chance that the committee consists of ladies majority
(a) \( \frac{7}{22} \)
(b) \( \frac{3}{22} \)
(c) \( \frac{5}{22} \)
(d) \( \frac{1}{22} \)
Answer: (d) \( \frac{1}{22} \)

Question. If the letters of the word NALGONDA are arranged in arbitrary order, the probability that the letter G, O, D appear in that order is
(a) \( \frac{1}{6} \)
(b) \( \frac{1}{24} \)
(c) \( \frac{1}{120} \)
(d) \( \frac{1}{10} \)
Answer: (a) \( \frac{1}{6} \)

Question. The adjoining figure gives the road plan of lanes connecting two parallel roads AB and FJ. A man walking on the road AB takes a turn at random to reach the road FJ. It is known that he reaches the road FJ from O by taking a straight line. The chance that he moves on a straight line from the road AB to the road FJ is.
(a) \( \frac{1}{5} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{3}{5} \)
(d) \( \frac{4}{5} \)
Answer: (a) \( \frac{1}{5} \)

Question. A is a set containing n elements. A subset \( P_{1} \) of A is chosen at random. The set A is reconstructed by replacing the elements of \( P_{1} \). A subset \( P_{2} \) is again chosen at random. The probability that \( P_{1} \cup P_{2} \) contains exactly one element, is
(a) \( \frac{3n}{4^{n}} \)
(b) \( \frac{3^{n}}{4^{n}} \)
(c) \( \frac{3}{4} \)
(d) \( \frac{3}{4^{n}} \)
Answer: (a) \( \frac{3n}{4^{n}} \)

Question. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{5} \)
(c) \( \frac{1}{10} \)
(d) \( \frac{1}{20} \)
Answer: (c) \( \frac{1}{10} \)