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Question Bank No: 2

1. A randon variable X has the following distribution
pro74 For the event E= ${X is prime number}$ and F= ${X < 4}, P (E \cup F) =$

a)		0.87
b)		0.77
c)		0.35
d)		0.50

2. Five horse are in a race. A person selects 2 of the horses at random and bets on them. The probability that he selected the winning horse is

a)		$\frac{4}{5}$
b)		$\frac{3}{5}$
c)		$\frac{1}{5}$
d)		$\frac{2}{5}$

3. A and B are events such that $P (A \cup B) = \frac{3}{4}, P (A \cap B) = \frac{1}{4}, P (\bar{A}) = \frac{2}{3}$ . Then $P (\bar{A} \cap B) =$

a)		$\frac{5}{12}$
b)		$\frac{3}{8}$
c)		$\frac{5}{8}$
d)		$\frac{1}{4}$

4. A problem in mathematics is given to three students and their respective probability of solving the problem are $\frac{1}{7}, \frac{7}{9}, \frac{1}{4}$ . The probability that the problem is solved is

a)		$\frac{3}{4}$
b)		$\frac{1}{2}$
c)		$\frac{2}{3}$
d)		$\frac{1}{3}$

5. A person goes to office either by ar, scooter, bus or train the probabilitites of which being $\frac{1}{7}, \frac{3}{7}, \frac{2}{7}, \frac{1}{7}$ respectively. The probability that he reaches office late, if he take car, scooter, bus or train is $\frac{2}{9}, \frac{1}{9}$ , $\frac{4}{9}, \frac{1}{9}$ respectively. If he reached office in time, the probability that he travelled by car is

a)		$\frac{1}{5}$
b)		$\frac{1}{9}$
c)		$\frac{2}{11}$
d)		$\frac{1}{7}$

6. A is targeting to B, B and C are targeting to A. The probability of hitting the target by A, B and C are $\frac{2}{3}, \frac{1}{2}, \frac{1}{3}$ respectively. If A is hit, the probability, that B hits the target and C does not, is

a)		$\frac{1}{3}$
b)		$\frac{1}{2}$
c)		$\frac{2}{3}$
d)		$\frac{3}{4}$

7. A box contain N coins, of which m are fair and the rest are biased. The probability of getting head when a fair coin is tossed is $\frac{1}{2}$ , while it $\frac{2}{3}$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shoes head and the second time it shows tail. The probability that the coin draen is fair is

a)		$\frac{5 m}{m + 8 N}$
b)		$\frac{3 m}{m + 8 N}$
c)		$\frac{75 m}{m + 8 N}$
d)		$\frac{9 m}{m + 8 N}$

8. An urn contain m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. If now a ball is drawn, the probability, that it is white, is

a)		$\frac{m + k}{m + n + k}$
b)		$\frac{n + k}{m + n + k}$
c)		$\frac{m}{m + n + k}$
d)		$\frac{m}{m + n}$

9. If p, q are chosen randomly with replacement from the set ${1, 2, 3......., 10}$ , the probability, that the roots of the equation $x^{2} + px + q = 0$ are real is

a)		$\frac{3}{5}$
b)		$\frac{31}{50}$
c)		$\frac{61}{100}$
d)		$\frac{29}{50}$

10. The probabilitities that a student passes in Mathematics, Physics and Chemistry are m,p and c respectively. Of these subjects, the srudent has 75% chance of passing atleast one, 50% chance of passing in atleast two and 40% chance of passing in exactly two. Then

a)		$p + m + c = \frac{19}{20}$
b)		$p + m + c = \frac{27}{20}$
c)		$p m c = \frac{2}{10}$
d)		$p m c = \frac{1}{4}$

11. There are 2 vans each having numbered seats, 3 in the front and 4 at the back. There are 3 girls and 9 boys to be seated in the vans. The probability of 3 girls sitting together in a back row on adjacent seats is

a)		$\frac{1}{13}$
b)		$\frac{1}{39}$
c)		$\frac{1}{65}$
d)		$\frac{1}{91}$

12. An unbaised coin is tossed. If the result is a head, a pair of unbaised dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ....12 is picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is

a)		$\frac{192}{401}$
b)		$\frac{193}{401}$
c)		$\frac{193}{792}$
d)		$\frac{17}{75}$

13. Numbers are selected at random, one at a time, from the two digit numbers 00, 01, 02,....99 with replacement. An event E occurs if and onlu if the product of the two digits of a selected number is 18. If 4 numbers are selected, the probability that the event E occurs atleast 3 times is

a)		$\frac{97}{5^{8}}$
b)		$\frac{79}{5^{6}}$
c)		$\frac{97}{5^{6}}$
d)		$\frac{79}{5^{8}}$

14. A lot contains 50 defective and 50 non defective bulbs. Two bulbs are drawn at random one at a time with replacement. The events A,B,C are defined as the first bulb is defective, the second bulb is non-defective, the two bulbs are both defective or non defective, respectively. Then

a)		A, B, C are pairwise independent
b)		A, B, C are pairwise not independent
c)		A, B, C are independent
d)		A, B, C are not dependent

15. A fair die is rolled. The probability that the first 1 occurs at the even throw is

a)		$\frac{1}{6}$
b)		$\frac{5}{11}$
c)		$\frac{6}{11}$
d)		$\frac{5}{36}$

16. If 3 distinct numbers are chosen randomly from the first 100 natural numbers. then the probability that all three of them are divisible by both 2 and 3 is

a)		$\frac{4}{55}$
b)		$\frac{4}{35}$
c)		$\frac{4}{33}$
d)		$\frac{4}{1155}$

17. Two numbers are selected randomly from the set S= ${1, 2, 3, 4, 5, 6}$ without replacement. The probability that minimum of the two number

a)		$\frac{1}{15}$
b)		$\frac{14}{15}$
c)		$\frac{1}{5}$
d)		$\frac{4}{5}$

18. If $P (B) = \frac{3}{4}$ , $P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C) =$

a)		$\frac{1}{12}$
b)		$\frac{1}{6}$
c)		$\frac{1}{15}$
d)		$\frac{1}{9}$

19. If the integers m and n are chosen at random between 1 and 100, then the probability that $7^{m} + 7^{n}$ is divisible by 5 is

a)		$\frac{1}{4}$
b)		$\frac{1}{7}$
c)		$\frac{1}{8}$
d)		$\frac{1}{49}$

20. If E and F are events with $P (E) \leq P (F)$ and $P (E \cap F) > 0,$ then

a)		occurence of E $\Rightarrow$ Occurence of F
b)		occurence of F $\Rightarrow$ Occurence of E
c)		non-occurence of E $\Rightarrow$ non-occurence of F
d)		none of these

21. If E and F are events such that $0 < P (F) < 1,$ then

a)		$P (\frac{E}{F}) + P (\frac{\bar{E}}{F}) = 1$
b)		$P (\frac{E}{F}) + P (\frac{E}{\bar{F}}) = 1$
c)		$P (\frac{\bar{E}}{F}) + P (\frac{E}{\bar{F}}) = 1$
d)		$P (\frac{E}{\bar{F}}) + P (\frac{\bar{E}}{\bar{F}}) = 0$

22. A fair coin is tossed repeatedly. If tail appears on first 4 tosses, then the probability of head appearing on the fifth toss is

a)		$\frac{1}{2}$
b)		$\frac{1}{32}$
c)		$\frac{31}{32}$
d)		$\frac{1}{5}$

23. Seven white balls and 3 black balls are randomly placed in a row. The probability that no two black balls are placed adjacently, is

a)		$\frac{1}{2}$
b)		$\frac{7}{15}$
c)		$\frac{2}{15}$
d)		$\frac{1}{3}$

24. There are 4 machines and it is known that exactly two of them are faulty. They are tested one by one in a random order tillboth the faulty machines are identified. Then the probability that only two tests are needed is

a)		$\frac{1}{3}$
b)		$\frac{1}{6}$
c)		$\frac{1}{2}$
d)		$\frac{1}{4}$

25. If from each of 3 boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn then the probability of drawing 2 white and 1 black ball is

a)		$\frac{13}{32}$
b)		$\frac{1}{4}$
c)		$\frac{1}{32}$
d)		$\frac{3}{16}$

26. Three numbers are chosen at random without replacement from ${1, 2, 3, ....., 10}$ .The probability tha the minimum of the chosen number is 3, or their maximum is 7 is

a)		$\frac{11}{20}$
b)		$\frac{7}{20}$
c)		$\frac{11}{20}$
d)		$\frac{7}{40}$

27. For three events A, B, C
P (exactly one of the events A or B occurs)
=P (exactly one of the events B or C occurs)
=P (exactly one of the events C or A occurs)
= p and P (all the three events occur simutaneously)= $p^{2},$ where $0 < p < \frac{1}{2}$ . Then the probability of atleast one of the three events A, B, C occurring is

a)		$\frac{3 p + 2 p^{2}}{2}$
b)		$\frac{p + 3 p^{2}}{4}$
c)		$\frac{p + 3 p^{2}}{2}$
d)		$\frac{3 p + 2 p^{2}}{4}$

28. 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)		$\frac{1}{20}$

29. The probability of India winning a test match against West aindies is $\frac{1}{2} .$ Assuming independence from match to match the probability that in a 5 match series India's second win occur at the third test is

a)		$\frac{1}{8}$
b)		$\frac{1}{4}$
c)		$\frac{1}{2}$
d)		$\frac{2}{3}$

30. Let $0 < P (A) < 1, 0 < P (B) < 1 and$ $P (A \cup B) = P (A) + P (B) - P (A) P (B) then$

a)		$P (\frac{B}{A}) = P (B) - P (A)$
b)		$P (\bar{A} - \bar{B}) = P (\bar{A}) - P (\bar{B})$
c)		$P (\bar{A \cup B}) = P (\bar{A}) \cdot P (\bar{B})$
d)		$P (\frac{A}{B}) = P (A) - P (B)$

31. If two events A and B are such that P( $\bar{A}$ )=0.3, P(B)=0.4 and P(A $\cap \bar{B}$ )=0.5, then $P (\frac{B}{(A \cup \bar{B})})$ =

a)		$\frac{1}{2}$
b)		$\frac{1}{3}$
c)		$\frac{2}{5}$
d)		$\frac{1}{4}$

32. An unbaised die with faces marked 1, 2, 3, 4, 5, 6 is rolled 4 times. Out of 4 face values obtained , the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

a)		$\frac{16}{81}$
b)		$\frac{1}{81}$
c)		$\frac{80}{81}$
d)		$\frac{65}{81}$

33. India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1, 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting atleast 7 points is

a)		0.8750
b)		0.0875
c)		0.0625
d)		0.0250

34. Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours yellow, red and blue appear in the first, second and third toss respectively is

a)		$\frac{1}{6}$
b)		$\frac{1}{12}$
c)		$\frac{1}{24}$
d)		$\frac{1}{36}$

35. Let A and B be two events such that P(A)=0.3 abd P(A $\cup$ B)=0.8; if A and B are independent events, then P(B)=

a)		0.5
b)		$\frac{3}{7}$
c)		$\frac{5}{7}$
d)		$\frac{2}{3}$

36. If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fail is

a)		0.56
b)		0.5
c)		0.44
d)		none of these

37. A pair of dice is rolled together till a sum of eithr 5 or 7 is obtained. The probability that 5 comes before 7 is

a)		$\frac{2}{5}$
b)		$\frac{2}{7}$
c)		$\frac{3}{7}$
d)		$\frac{3}{5}$

38. One hundred coins, each with probability p of showing up heads are tossed once. If $0 < p < 1$ and the probability of heads showing on 50 coins is equal to that of heads showing 51 coins, then p=

a)		$\frac{1}{2}$
b)		$\frac{49}{101}$
c)		$\frac{50}{101}$
d)		$\frac{51}{101}$

39. Urn A contains 6 red and 4 white balls and urn B contians 4 red and 6 white balls. One ball is drawn at random from urn A and placed in urn B. Then a ball is drawn urn B and placed in urn A. Now if one ball is drawn from urn A, the probability that it is red, is

a)		$\frac{6}{11}$
b)		$\frac{17}{50}$
c)		$\frac{16}{55}$
d)		$\frac{32}{55}$

40. An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white it is not replaced, otherwise it is replaced along with another ball of the same colour. The process is repeated. The probability that the third ball drawn is black is

a)		$\frac{11}{15}$
b)		$\frac{23}{30}$
c)		$\frac{22}{25}$
d)		none of these

41. A man takes a step forward with probability 0.4 and backwards with probability 0.6. The probability that at the end of 11 steps he is one step away from the starting point is

a)		420 ${(0.24)}^{5}$
b)		420 ${(0.24)}^{6}$
c)		462 ${(0.24)}^{5}$
d)		462 ${(0.24)}^{6}$

42. The probability that atleast one of the events A and B occurs in 0.6. If A and B occur simultaneously with probability 0.2 then $P (\bar{A}) + P (\bar{B}) =$

a)		0.4
b)		0.8
c)		1.2
d)		none

43. A student appears for tests I, II and III. The student is successful if he passes either in test I and II or test I and III. The probabilities of the student passing in tests I, II, III are p, q, $\frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2}$ , then

a)		p= q=1
b)		p=q= $\frac{1}{2}$
c)		p=1, q=1
d)		p=1, q= $\frac{1}{2}$

44. If $\frac{1 + 3 p}{3}$ , $\frac{1 - p}{4}$ and $\frac{1 - 2 p}{2}$ are the probabilities of the three mutually exclusive events, then p $\in$

a)		$[0, 1]$
b)		$[0, \frac{1}{2}]$
c)		$[\frac{1}{3}, 1]$
d)		$[\frac{1}{3}, \frac{1}{2}]$

45. A box contain 100 tickets numbered 1 to 100. Two tickets are chosen at random. It is given that the maximum number on them is not more than 10. The minimum number on them is 5 with probability

a)		$\frac{4}{45}$
b)		$\frac{1}{9}$
c)		$\frac{8}{45}$
d)		$\frac{1}{18}$

46. In a multiple-choice question there are 4 alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random. If he is allowed upt 3 chances to answer the question, the probability that he will get marks in the question is

a)		$\frac{1}{3}$
b)		$\frac{1}{4}$
c)		$\frac{1}{5}$
d)		none of these

47. Three identical dice are rolled once. The probability that the same number will appear on each of them is

a)		$\frac{1}{16}$
b)		$\frac{1}{36}$
c)		$\frac{1}{18}$
d)		$\frac{3}{28}$

48. A and B are two independent events. The probability that both A and B occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$ . The probability of occurence of A is

a)		$\frac{1}{2}$
b)		$\frac{1}{5}$
c)		$\frac{1}{4}$
d)		$\frac{1}{6}$

49. If the letters of the word ASSASSIN are written down at random in a row, the probability that no two S's occur together is

a)		$\frac{1}{35}$
b)		$\frac{1}{21}$
c)		$\frac{1}{14}$
d)		$\frac{1}{28}$

50. Seven coupons are selected at random one at a time with replacement from 15 coupons numbered 1 to 15. The probability that the largest number appearing on a selected coupon is 9, is

a)		${(\frac{9}{16})}^{6}$
b)		${(\frac{8}{15})}^{7}$
c)		${(\frac{3}{5})}^{7}$
d)		none of these