Classical Definition of Probability

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JEE Main 2015

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is

220 (1/3)^{12}

(B)

22 (1/3)^{11}

(C)

\frac{55}{3} (2/3)^{11}

(D)

55 (2/3)^{10}

JEE Main 2010

LEVELJEE Main

Four numbers are chosen at random (without replacement) from the set ${1, 2, 3, ..., 20}$ . Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $1/85$ . Statement-2: If the four chosen numbers form an AP, then the set of all possible values of common difference is ${\pm 1, \pm 2, \pm 3, \pm 4, \pm 5}$ .

(A)

Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1

(B)

Statement-1 is true, Statement-2 is false

(C)

Statement-1 is false, Statement-2 is true

(D)

Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1

JEE Main 2010

LEVELBoard

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is

A die is thrown. Let $A$ be the event that the number obtained is greater than 3. Let $B$ be the event that the number obtained is less than 5. Then $P (A \cup B)$ is

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

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

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

Four fair dice $D_{1}, D_{2}, D_{3}$ and $D_{4}$ ; each having six faces numbered 1, 2, 3, 4, 5 and 6 are rolled simultaneously. The probability that $D_{4}$ shows a number appearing on one of $D_{1}, D_{2}$ and $D_{3}$ is

If three 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

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

If the integers $m$ and $n$ are chosen at random from 1 to 100, then the probability that a number of the form $7^{m} + 7^{n}$ is divisible by 5 equals

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

JEE Advanced 1998

LEVELJEE Main

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

(A)

occurrence of

E \Rightarrow

occurrence of

F

(B)

occurrence of

F \Rightarrow

occurrence of

E

(C)

non-occurrence of

E \Rightarrow

non-occurrence of

F

(D)

none of the above implications holds

JEE Advanced 1998

LEVELBoard

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

(A)

1/2

(B)

1/32

(C)

31/32

(D)

1/5

JEE Advanced 1998

LEVELJEE Main

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

(A)

1/2

(B)

7/15

(C)

2/15

(D)

1/3

JEE Advanced 1995

LEVELJEE Main

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

(A)

1/2

(B)

1/5

(C)

1/10

(D)

1/20

JEE Advanced 1984

LEVELBoard

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

(A)

1/6

(B)

1/36

(C)

1/18

(D)

3/28

JEE Advanced 1983

LEVELJEE Main

Fifteen coupons are numbered $1, 2, \dots, 15$ , 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 $9$ , is

(A)

(\frac{9}{16})^{6}

(B)

(\frac{8}{15})^{7}

(C)

(\frac{3}{5})^{7}

An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in this list?

JEE Advanced 1999

LEVELJEE Main

Eight players $P_{1}, P_{2}, \dots, P_{8}$ play a knock-out tournament. It is known that whenever the players $P_{i}$ and $P_{j}$ play, the player $P_{i}$ will win if $i < j$ . Assuming that the players are paired at random in each round, what is the probability that the player $P_{4}$ reaches the final?

JEE Advanced 1990

LEVELJEE Main

$A$ is a set containing $n$ elements. A subset $P$ of $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P$ . A subset $Q$ of $A$ is again chosen at random. Find the probability that $P$ and $Q$ have no common elements.

JEE Advanced 1988

LEVELJEE Main

A box contains 2 fifty paise coins, 5 twenty five paise coins and a certain fixed number $N (N \geq 2)$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these 5 coins is less than one rupee and fifty paise.

JEE Advanced 1986

LEVELJEE Main

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

JEE Advanced 1985

LEVELJEE Main

In a multiple-choice question there are four 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 upto three chances to answer the questions, find the probability that he will get marks in the questions.

JEE Advanced 1982

LEVELJEE Main

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of determinant chosen is positive is .........

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A∪B) is

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

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

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

If three 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two numbers are selected randomly from the set S={1,2,3,4,5,6} without replacement one by one. The probability that minimum of the two numbers is less than 4 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7m+7n is divisible by 5 equals

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If E and F are events with P(E)≤P(F) and P(E∩F)>0, then

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

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 31+3p​,41−p​ and 21−2p​ are the probabilities of three mutually exclusive events, then the set of all values of p is .........

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of determinant chosen is positive is .........

A die is thrown. Let $A$ be the event that the number obtained is greater than 3. Let $B$ be the event that the number obtained is less than 5. Then $P (A \cup B)$ is

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

If the integers $m$ and $n$ are chosen at random from 1 to 100, then the probability that a number of the form $7^{m} + 7^{n}$ is divisible by 5 equals

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

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