Choose the correct alternative. Only one choice is correct.
1. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these vertices is equilateral is :
(A) 1/2 (B) 1/3 (C) 1/10 (D) 1/20
2. The probability that the length of a randomly chosen chord of a circle lies between 2/3 and 5/6 of its diameter is :
(A) 1 / 4 (B) 5 / 12 (C) 1 / 16 (D) 5 / 16
3. The coefficients of a quadratic equation ax2bx c 0
a b c
are chosen from first three prime numbers, the probability that roots of the equation are real is :(A) 1/3 (B) 2/3 (C) 1/4 (D) 3/4
4. Let a die is loaded in such a way that prime number faces are twice as likely to occur as a non prime number faces. The probability that an odd number will be show up when the die is tossed is :
(A) 1/3 (B) 2/3 (C) 4/9 (D) 5/9
5. A random variable X has the following probability distribution then, is equal to :
X P(X = x) X P (X = x) 0 1 2 3 4 3 5 7 9 5 6 7 8 11 13 15 17 (A) 1/81 (B) 2/81 (C) 5/81 (D) 7/81
6. Two dice are rolled one after another. The probability that the number on the first is less than or equal to the number on the second is :
(A) 5/12 (B) 7/12 (C) 5/18 (D) 13/18
7. Consider the circuit, if the probability that each switch is closed is p, then find the probability of current flowing through AB.
(A) p2 + p (B) p2 p1 (C) p3 p (D) p2 p 1
8. A rifle man firing at a distant target and has only 10% chance of hitting it. The minimum number of rounds he must fire in order to have 50% chance of hitting it atleast once is :
B A
a b
(A) 2/5 (B) 3/5 (C) 4/5 (D) 1/7
10. A boy is throwing stones at a target. The probability of hitting the target at any trial is 1/2. The probability of hitting target 5th time at the 10th thrown is : (A) 10 5 2 (B) 9 63 2 (C) 10 5 10 2 C (D) 10 4 10 2 C
Choose the correct alternative. Only one choice is correct. 1. An ellipse of eccentricity 2 2
3 is inscribed in a circle and a point within the circle is chosen at random. The probability that this point lies outside the ellipse is :
(A) 1/3 (B) 2/3 (C) 1/9 (D) 2/9
2. A square is inscribed in a circle. If p1 is the probability that a randomly chosen point of the circle lies within the square and p2 is the probability that the point lies outside the square, then :
(A) p1 p2 (B) p1 p2 (C) p1 > p2 & 12 22
1 3
p p (D) None of these
3. A die is rolled three times, the probability of getting a large number than the previous number is :
(A) 1/54 (B) 5/54 (C) 5/108 (D) 13/108
4. Two persons A and B agree to meet at a place between 11 to 12 noon. The first one to arrive waits for 20 min and then leave. If the time of their arrival be independent and at random, then the probability that A and B meet is :
(A) 1/3 (B) 2/3 (C) 4/9 (D) 5/9
5. In a game called “odd man out”, n(n > 2) persons toss a coin to determine who will buy refreshments for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. If the probability that there is a loser in any game is 1/2, then the value of n is :
(A) 4 (B) 7 (C) 8 (D) 11
6. The probabilities that a student will obtain grades A, B, C or D are 0.30, 0.35, 0.20 and 0.15 respectively. The probability that he will receive atleast C grade, is :
(A) 0.65 (B) 0.85 (C) 0.80 (D) 0.20
7. 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 fails, is :
(A) 0.38 (B) 0.44 (C) 0.50 (D) 0.94
8. There are 5 duplicate and 10 original items in an automobile shop and 3 items are bought at random by a customer. The probability that none of the items is duplicate, is :
(A) 20/91 (B) 22/91 (C) 24/91 (D) 89/91
9. A man is known to speak truth in 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is :
(A) 1/6 (B) 1/8 (C) 3/4 (D) 3/8
10. 10 bulbs out of a sample of 100 bulbs manufactured by a company are defective. The probability that 3 out of 4 bulbs, bought by a customer will not be defective, is :
Choose the correct alternative. Only one choice is correct.
1. Two small squares on a chess board are chosen at random. Probability that they have a common side is :
(A) 1/3 (B) 1/9 (C) 1/18 (D) 5/18
2. A sum of money is rounded off to the nearest rupee, the probability that the rounded off error is atleast ten paise is:
(A) 1/100 (B) 9/100 (C) 49/100 (D) 81/100
3. A dice is thrown (2n + 1) times. The probability that faces with even numbers appear odd number of times is :
(A) 2 1 2 3 n n (B) 1 2 1 n n (C) 2 1 n n (D) 1 2
4. For n independent events Ai’s, P(Ai) = 1 / (1 + i), i = 1, 2, …... n. The probability that atleast one of the events occurs is :
(A) I/n (B) 1/(n + 1) (C) n/(n + 1) (D) None of these
5. The probability that a teacher will give an unannounced test during any class meeting is 1/5. If a student is absent twice, the probability that he will miss atleast one test, is :
(A) 7/25 (B) 9/25 (C) 16/25 (D) 24/25
6. A box contains 15 transistors, 5 of which are defective. An inspector takes out one transistor at random, examines it for defects, and replaces it. After it has been replaced another inspector does the same thing, and then so does a third inspector. The probability that atleast one of the inspectors find a defective transistor, is equal to :
(A) 1/27 (B) 8/27 (C) 19/27 (D) 26/27
7. Three letters are written to three different persons and addresses on the three envelopes are also written. Without looking at the addresses, the letters are kept in these envelopers. The probability that all the letters are not placed into their envelopes is :
(A) 1/2 (B) 1/3 (C) 1/6 (D) 5/6
8. A bag contains 7 red and 2 white balls and another bag contains 5 red and 4 white balls. Two balls are drawn, one from each bag. The probability that both the balls are white, is :
(A) 2/9 (B) 2/3 (C) 8/81 (D) 35/81
9. A bag contains 5 red, 3 white and 2 black balls. If a ball is picked at random, the probability that it is red, is :
(A) 1/5 (B) 1/2 (C) 3/10 (D) 9/10
10. A single letter is selected at random from the word ‘PROBABILITY’. The probability that it is a vowel, is :
Choose the correct alternative. Only one choice is correct.
1. A piece of wire of length 4L is bent at random, to form a rectangle. The probability that its area is at most
2 4 L is : (A) 1 4 (B) 3 2 (C) 2 3 2 (D) 4 3
2. Fifteen coupons are numbered 1, 2, 3, …… , 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is atmost 9, is :
(A)
1 15/
7 (B)
3 5/
7 (C)
8 15/
7 (D) None of these3. Three athletes A, B, and C participate in a race. Both A and B have the same probability of winning the race and each is twice as likely to win as C. The probability that B or C wins the race is :
(A) 2/3 (B) 3/5 (C) 3/4 (D) 13/25
4. A number is chosen at random from among the first 30 natural numbers. The probability of the number chosen being a prime, is :
(A) 1/3 (B) 3/10 (C) 1/30 (D) 11/30
5. Out of 13 applicants for a job, there are 8 men and 5 women. It is desired to select 2 persons for the job. The probability that atleast one of the selected persons will be a woman, is :
(A) 5/13 (B) 10/13 (C) 14/39 (D) 25/39
6. Two athletes A and B participate in a race along with other athletes. If the chance of A winning the race is 1/6 and that of B winning the same race is 1/8, then the chance that neither wins the race, is :
(A) 1/4 (B) 7/24 (C) 17/24 (D) 35/48
7. A mapping is selected at random from the set of all mapping of the set A = (1, 2, ……., n) into itself. The probability that the mapping selected is bijective is :
(A) 1 n (B) 1 n n (C) 2n n (D) n n n
8. The probability of guessing correctly atleast 8 out of 10 answers on a true - false examination, is :
(A) 7/64 (B) 7/128 (C) 45/1024 (D) 175/1024
9. The probability that the 13th day of a randomly chosen month is a second Saturday, is :
(A) 1/7 (B) 1/12 (C) 1/84 (D) 19/84
10. A purse contains 4 copper and 3 silver coins, and a second purse contains 6 copper and 2 silver coins. A coin is taken our from any purse, the probability that it is a copper coin, is :
Choose the correct alternative. Only one choice is correct.
1. The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that second can be subtracted from one first without borrowing is :
(A) 6 1 2 (B) 6 9 20 (C) 6 11 20 (D) 6 3 20
2. The probability that a man aged x years will die in a year is p. The probability that out of n men M1, M2, ...., Mn, each aged x, M1 will die and be the first to die, is :
(A) 1 / n2 (B) 1
1 p
n (C)
2 1 1 1 n n p (D) 1 1
1 p
n n 3. n letters are written to n different persons and addresses on the n envelopes are also written. If the letters are placed in the
envelopes at random, the probability that atleast one letter is not placed in the right envelope, is :
(A) 1 1 n (B) 1 1 2n (C) 2 1 1 n (D) 1 1 n
4. Three players A, B, C in this order, cut a pack of cards, and the whole pack is reshuffled after each cut. If the winner is one who first draws a diamond, then C’s chance of winning is :
(A) 9/28 (B) 9/37 (C) 9/64 (D) 27/64
5. 6 girls and 6 boys sit together randomly in a row, the probability that no two boys sit together, is :
(A) 6 5 11 (B) 6 6 11 (C) 6 7 2 11 (D) 5 7 2 11
6. An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four times, Out of the four 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) 1/81 (B) 16/81 (C) 65/81 (D) 80/81
7. A box contains cards numbered 1 to 100. A card is drawn at random from the box. The probability of drawing a number which is a square, is :
(A) 1/5 (B) 2/5 (C) 1/10 (D) None of these
8. A three-digit number is selected at random from the set of all three - digit numbers. The probability that the number selected has all the three digits same, is :
(A) 1/9 (B) 1/10 (C) 1/50 (D) 1/100
9. If the mean and variance of a binomial variate X are 7/3 and 14/9 respectively. Then probability that X takes value 6 or 7 is equal to :
(A) 1/729 (B) 5/729 (C) 7/729 (D) 13/729
10. One ticket is selected at random from 100 tickets numbered 00, 01, 02, ….., 99. Suppose X and Y are the sum and product of the digit sound on the ticket P(X = 7 / Y = 0) is given by :
Choose the correct alternative. Only one choice is correct.
1. An integer is chosen at random from the numbers 1, 2, ….. 25. The probability that the chosen number is divisible by 3 or 4, is :
(A) 2/25 (B) 11/25 (C) 12/25 (D) 14/25
2. Two player A and B throw a die alternately for a prize of Rs. 11, which is to be won by a player who first throws a six. If A starts the game, their respective expectations are:
(A) Rs.6 ; Rs.5 (B) Rs.7; Rs. 4 (C) Rs.5.50 ; Rs.5.50 (D) Rs, 5.75 ; Rs. 5.25
3. The probability of a man hitting a target in one fire is 1/4. Atleast n times, he must fire at the target that the chances of hitting the target atleast once will exceed 2/3, then n is :
(A) 2 (B) 8 (C) 6 (D) 4
4. A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw. A begins the game. If the probability of A wining (that is drawing a white ball) is twice the probability of B winning (i.e. drawing a black ball), then the ratio a : b is equal to :
(A) 1 : 2 (B) 2 : 1 (C) 1 : 1 (D) None of these
5. Let X be a set containing n elements. Two subsets A and B of X are chosen at random, the probability that ABX is : (A) 2nC /n 22n (B) 1/2nCn (C) 1 3 5 ....
2n1
/2nn (D)
3 4/
n 6. A natural number x is chosen at random from the first 100 natural numbers. The probability that x 100 50x
is :
(A) 1/10 (B) 11/50 (C) 1/20 (D) 11/20
7. 10% bulbs manufactured by a company are defective. The probability that out of a sample of 5 bulbs, none is defective, is : (A)
1 2/ 5 (B)
1 10/
5 (C)
9 10/
(D)
9 10/
58. Let A = {1, 3, 5, 7, 9) and B = {2, 4, 6, 8}. An element (a, b) of their cartesian product A Bis chosen at random. The probability that (a + b) = 9, is :
(A) 1/5 (B) 2/5 (C) 3/5 (D) 4/5
9. If A and B are any two events, then the probability that exactly one of them occurs, is : (A) P A
B
P AB
(B) P A
B
P AB
(C) P A
P B
P A
B
(D) P A
P B
2P A
B
10. If the integers and are chosen at random between 1 to 100, then the probability that a number of the form 7 7 is divisible by 5, is :
Choose the correct alternative. Only one choice is correct.
1. In a college, 20% students fail in mathematic, 25% in Physics, and 12% in both subjects. A student of this college is selected at random. The probability that this student who has failed in Mathematics would have failed in physics too, is :
(A) 1/20 (B) 3/25 (C) 12/25 (D) 3/5
2. One mapping is selected at random from all the mapping of the set A = {1, 2, 3, . . . . ., n} into itself. The probability that the mapping selected is one to one is given by :
(A) 1 n n (B) 1 n (C)
1 1 n n n (D) None of these3. If X and Y are independent binomial variates B(5, 1/2) and B (7, 1/2), then P(X + Y = 3) is :
(A) 55/1024 (B) 55/4098 (C) 55/2048 (D) None of these
4. Eight coins are tossed at a time, the probability of getting atleast 6 heads up is :
(A) 7/64 (B) 57/64 (C) 37/256 (D) 229/256
5. The probability that atleast one of the events A and B occur is 0.6. If A and B occur simultaneously with probability 0.2, then P A
P B , where Aand B are complements of A and B respective, is equal to :(A) 0.4 (B) 0.8 (C) 1.2 (D) 1.4
6. A speaks truth in
60%
cases and B speaks truth in 70% cases. The probability that they will say the same thing while describing a single event is :(A) 0.56 (B) 0.54 (C) 0.38 (D) 0.94
7. A box contains tickets numbered 1 to 20, 3 tickets are drawn from the box with replacement. The probability that the largest number on the tickets is 7, is ;
(A) 7/20 (B) 1
7 20/
3 (C) 2/10 (D) None of these8. Seven digits from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a random order. The probability that this seven digit number is divisible by 9 is :
(A) 2/9 (B) 1/5 (C) 1/3 (D) 1/9
9. A bag contains four tickets numbered 00, 01, 10, 11. Four tickets are chosen at random with replacement, the probability that sum of the numbers on the tickets is 23, is :
(A) 3/32 (B) 1/64 (C) 5/256 (D) 1/2
10. Two integers x and y are chosen, without replacement, at random from the set
x |0 x 10 xis an integer
the probability that 5| x y| is :
Choose the correct alternative. Only one choice is correct.
1. The probabilities of two events A and B are 0.3 and 0.6 respectively. The probability that both A and B occur simultaneously is 0.18. Then, the probability that neither A nor B occurs is :
(A) 0.10 (B) 0.28 (C) 0.42 (D) 0.72
2. Of the 25 questions in a unit, a student has worked out only 20. In a sessional test of that unit, two questions were asked by the teacher. The probability that the student can solve both the questions correctly, is :
(A) 8/25 (B) 17/25 (C) 9/10 (D) 19/30
3. Dialing a telephone number, a man forgot the last two digits and remembering only that they are different, dialed them at random. The probability of the number being dialled correctly is :
(A) 1/2 (B) 1/45 (C) 1/72 (D) 1/90
4. If two events A and B are such that P A
0 andP B
1 then P A / B
is equal to : (A) 1P A / B
(B) 1P A / B
(C)
1 P A B P B (D)
P A P B5. A three digit number, which is multiple of 11, is chosen at random. The probability that the number so chosen is also a multiple of 9 is equal to : (A) 1/9 (B) 2/9 (C) 1/100 (D) 9/100 6. If 1 4 1 1 2 4 2 2 p p p
are probabilities of three mutually exclusive events, then :
(A) 1 1 3 p 2 (B) 1 2 3 p 3 (C) 1 1 6 p 2 (D) None of these
7. Two numbers a and b are chosen at random from the set {1, 2, 3 …., 30}. The probability that a2b2 is divisible by 3 is :
(A) 3/29 (B) 4/29 (C) 5/29 (D) None of these
8. A fair die is thrown until a score of less than five points is obtained. The probability of obtaining less than three points on the last throw is :
(A) 3/4 (B) 4/5 (C) 5/6 (D) 1/2
9. A wire of length is cut into three pieces. What is the probability that the three pieces form a triangle ?
(A) 1/2 (B) 1/4 (C) 2/3 (D) None of these
10. A natural number is selected at random from the setX
x |1 x 100
where x is an integer. The probability that the number satisfies the inequation x2– 13x 30, is :Choose the correct alternative. Only one choice is correct.
1. Suppose X is a binomial variate B (5, p) and P(X = 2) = P(X = 3), then p is equal to :
(A) 1/2 (B) 1/3 (C) 1/4 (D) 1/5
2. The adjoining figure gives the road plan of lines connecting two parallel roads AB and A1B1. A man walking on the road AB takes turn at random to reach the rod A1B1. It is known that he reaches the road A1B1 from O by taking a straight line. The chance that he moves on a straight line from the road AB to the road A1B1 is :
(A) 0.25 (B) 0.04
(C) 0.2
(D) None of these
3. If a 20 0 , then the probability that the graph of the function y = 16x2 + 8(a + 5) x7a5 is strictly above the X-axis is :
(A) 1/2 (B) 1/17 (C) 17/20 (D) None of these
4. Given that x 0 1 and y 0 1. Let A be the event of (x, y) satisfying y2 xand B be the event of (x, y) satisfying x2 ,y
then :
(A)
13
P AB (B) A, B are exclusive
(C) P A
P B
(D) P B
P A
5. A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained, the probability that 5 comes before 7 is :
(A) 0.2 (B) 0.3 (C) 0.4 (D) 0.5
6. A five digit number is chosen at random. The probability that all the digits are distinct and digits at odd place are odd and digits at even places are even is :
(A) 1/25 (B) 25/567 (C) 1/37 (D) 1/74
7. x1, x2, x3, …….. x50 are fifty real numbers such thatxr xr1 for r = 1, 2, 3, ……, 49. Five numbers out of these are picked up at
random. The probability that the five numbers have x20 as the middle number is :
(A) 20 30 2 2 20 5 C C C (B) 30 19 2 2 50 5 C C C (C) 19 30 2 3 50 5 C C C (D) None of these
8. A die is thrown 2n1 times, nN. The probability that faces with even numbers show odd number of times is :
(A) 2 1 2 3 n n (B) 1 less than
2 (C) greater than 1/2 (D) None of these
A E D C B O D1 E1 A1 B1 C1
Probability 0.1 0.32 0.31 0.15 0.05 0.17
The dice is thrown and it is known that either the face number 1 or 2 will appear. Then, the probability of the face number 1 to appear is :
Choose the correct alternative. Only one choice is correct.
1. Two distinct numbers are selected at random from the first twelve natural numbers. The probability that the sum will be divisible by 3 is:
(A) 1/3 (B) 23/66 (C) 1/2 (D) None of these
2. The probability that out of 10 persons, all born in April, at least two have the same birthday is :
(A)
30 10 10 30 C (B) 30 10 1 30 C (C)
10 30 10 10 30 30 C (D) None of these3. A second order determinant is written down at random using the numbers 1 as elements. The probability that the value of the1 determinant is non zero is :
(A) 1/2 (B) 3/8 (C) 5/8 (D) 1/3
4. A natural number is selected from 1 to 1000 at random, then the probability that a particular non-zero digit appears atmost once is :
(A) 3/250 (B) 143/250 (C) 243/250 (D) 7/250
5. If A and B are events at the same experiments with P(A) = 0.2, P(B) = 0.5, then maximum value ofP A
B
is:(B) 1/4 (B) 1/2 (C) 1/8 (D) 1/16
6. A bag contains 14 balls of two colors, the number of balls of each color being the same. 7 balls are drawn at random one by one. The ball in hand is returned to the bag before each new drawn. If the probability that atleast 3 balls of each colour are drawn is p, then : (A) 1 2 p (B) 1 2 p (C) p < 1 (D) 1 2 p
7. All the spades are taken out from a pack of cards. From these cards; cards are drawn one by one without replacement till the ace of spades comes. The probability that the ace comes in the 4th draw is :
(A) 1/13 (B) 12/13 (C) 4/13 (D) None of these
8. If X follows a binomial distribution with parameters n = 8 and p = 1/2, then p | x
4|2
is equal to :(A) 121/128 (B) 119/128 (C) 117/128 (D) 115/128
9. Two numbers b and c are chosen at random (with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9). The probability that
2 0
x bx for all x Rc is :
(A) 17/123 (B) 32/81 (C) 82/125 (D) 45/143
10. Let A, B, C be three mutually independent events. Consider the two statements S1 and S2.
S1 : Aand B are independentC S2 : Aand BC are independent, then
(A) Both S1 and S2 are true (B) Only S1 is true
FILL IN THE BLANKS :
1. A natural number x is chosen at random from the first one hundred natural numbers. The probability that
20
40 0 30 x x x is …………...2. A number is chosen at random from the numbers 10 to 99. By seeing the number a man will laugh if product of the digits is 12. If he chose three numbers with replacement, then the probability that he will laugh atleast once is ……….
3. Sixteen players P1, P2, ……., P16 play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the players are of equal strength, the probability that exactly one of the two players P1 and P2 is among the eight winners is ………….
4. A is a set containing 10 elements. A subset P of A is chosen at random and the set A reconstructed by replacing the elements of P.
Another subset Q of A is now chosen at random. Then, the probability random and of PQ A is……..
5. A biased coin with probability p 0 p1 of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 1/6, then p = ……….
6. If P A
B
P A
B
the relation between P(A) and P(B) is ………7. A card is drawn at random from a pack of 52 cards. The chance that it is either an ace or a heart is ………
8. There are 10 persons among whom A and B, are made to stand in a row in random order. The probability that there is exactly one person between A and B is ……..
9. 5 girls and 10 boys sit at random in a row having 15 chairs numbered as 1 to 15, then the probability that end seats are occupied by the girls and between any two girls odd number of boys sit is ……..
10. Consider the cartesian plane R2, and let X denote the subset of points for which both coordinates are integers. A coin of diameter 1/2 is tossed randomly onto the plane, then the probability p that the coin covers a point of X, is ………..
FILL IN THE BLANKS :
1. A letter is selected from each of the words ‘MATHEMATICS’ and ‘STATISTICS’ the probability that the same letters are selected is …………
2. A bag contains R identical red and G identical green balls. 2k balls are selected at random from the bag. The probability that there are more red ball than green balls is ………
3. A tells truth with probability p1 and B with probability p2. If they make the same statement, the probability that their statement is true is ………
4. A five digit number is chosen at random. The probability that all the digits are distinct and digits at odd places are odd and digits at even places are even (no digit being repeated) is ………..
5. The mean and variance of a binomial variable X are 2 and 1 respectively. Then the probability that X takes values greater than 1 is . . . .
6. There are 10 pairs of shoes in a cupboard, from which 4 shoes are picked at random. The probability that there is atleast one pair, is ………….
7. Only four students A, B, C, D appear in an examination. The probability of a coming first is 2 times that of B, probability of A coming first is 3 times that of C and probability of C coming first is 4 times that of D. The probability that either A or D comes first is ………
8. Five persons entered the lift cabin on the ground floor of 8 floor house. Suppose that each of them, independently and with equal probability, can leave the cabin at any floor beginning with the first, then the probability of all five persons leaving at different floors is ………..
9. A can hit a target 4 times in 5 shots, B can hit the same 3 times in 4 shots and C twice in 3 shots. If they fire one each, the
probability that at least two of them hit the target is ………
