Probability Questions - JEE

Two players $P_1$ and $P_2$ play a game. Each player rolls a die once. If $x>y$: $P_1$ scores 5, $P_2$ scores 0. If $x=y$: each scores 2. If $x<y$: $P_1$ scores 0, $P_2$ scores 5. Match: I) $P(X_2\ge Y_2)$; II) $P(X_2>Y_2)$; III) $P(X_3=Y_3)$; IV) $P(X_3>Y_3)$ with P) 3/8, Q) 11/16, R) 5/16, S) 355/864, T) 77/432.

A pair of dice is rolled together till a sum of either 5 or 7 is obtained. Find the probability that 5 comes before 7.

Let $S = \{w_1, w_2, \ldots\}$ be the sample space associated to a random experiment. Let $P(w_n) = \frac{P(w_{n-1})}{2}$, $n \geq 2$. Let $A = \{2k+3\ell; k, \ell \in \mathbb{N}\}$ and $B = \{w_n; n \in A\}$. Then P(B) is equal to

A bag contains n + 1 coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is 7/12, then find the value of n.

For any three events $A$, $B$ and $C$ defined on the sample space $R$ such that $B \subset C$ and $P(A) \neq 0$, $P(B/A) \leq P(C/A)$.

Let $S=\{2,3,5,7,11\}$. $A$ and $B$ are two matrices of order 2 each with distinct elements from $S$. Probability that matrix $AB$ has at least one odd entry is

A father has 3 children with at least one boy. The probability that he has 2 boys and 1 girl is

The probability of happening an event A in one trial is 0.4. Find the probability that the event A happens at least once in three independent trials.

An experiment succeeds twice as often as it fails. The probability of at least five successes in the six trials of this experiment is

If 3 identical cards are coloured both the sides such that the first card are colored red, both sides of the second card are colored black, and one side of the third card is colored red and the other side black. The 3 cards are mixed up, and 1 card is randomly selected and put down on the ground. If the upper side of the chosen card is colored black, what is the probability that the other side is colored red?

An event $X$ can take place in conjuction with any one of the mutually exclusive and exhaustive events $A$, $B$ and $C$. If $A$, $B$, $C$ are equiprobable and the probability of $X$ is 5/12, and the probability of $X$ taking place when $A$ has happened is 3/8, while it is 1/4 when $B$ has taken place, then the probability of $X$ taking place in conjuction with $C$ is

In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?

A problem in mathematics is given to three students $A$, $B$, $C$ and their respective probability of solving the problem is 1/2, 1/3, and 1/4. Probability that the problem is solved is

There are 3 bags which are known to contain 2 white and 3 black, 4 white and 1 black, and 3 white and 7 black balls, respectively. A ball is drawn at random from one of the bags and found to be a black ball. Then the probability that it was drawn from the bag containing the most black balls is

One mapping is selected at random from all mappings of the set $S = \{1, 2, 3, \ldots, n\}$ into itself. If the probability that the mapping is one-one is 3/32, then the value of $n$ is

An unbiased coin is tossed 6 times. The probability that third head appears on the sixth trial is

Two positive real numbers $x$ and $y$ satisfying $x\leq1$ and $y\leq1$ are chosen at random. The probability that $x+y\leq1$, given that $x^2+y^2\geq\frac{1}{4}$, is

Let $A$, $B$, $C$ be three mutually independent events. Consider the two statements $S_1$ and $S_2$.$S_1$: $A$ and $B \cup C$ are independent.$S_2$: $A$ and $B \cap C$ are independent.Then

Question nos. 646 to 648Let $X = \{1, 2, 3, \ldots, 10\}$. $A$, $B$, $C$ are three sets such that $A \subseteq X$, $B \subseteq X$ and $C \subseteq X$.Column-1: Contains types of three subsets of $X$.Column-2: Contains number of ways of selecting three subsets of $X$ according to column-1.Column-3: Contains conditional probabilities $P\!\left(\dfrac{E}{E_1}\right)$ or $P\!\left(\dfrac{E}{E_2}\right)$ where$E$: Selecting three subsets of $X$ according to column-1$E_1$: Selecting three subsets of $X$ such that $n(A \cap B) = 5$$E_2$: Selecting three subsets of $X$ such that $n(A \cup B) = 5$.Column-1 Column-2 Column-3(I) $A \cap B \cap C \supseteq \{2,3,4,5,6\}$ and $A = B = C$ (i) 32 (P) $P\!\left(\dfrac{E}{E_1}\right) = 0$(II) $A \cup B \cup C = \{3,4,5\}$ (ii) 242 (Q) $P\!\left(\dfrac{E}{E_1}\right) = \dfrac{1}{{}^{10}C_5 \cdot 12^5}$(III) $A \cap B \cap C = \{3,4,5,6,7\}$ and $A = B \neq C$ (iii) 243 (R) $P\!\left(\dfrac{E}{E_2}\right) = \dfrac{31}{{}^{10}C_5 \cdot 12^5}$(IV) $A \cup B \cup C = \{6,7,8,9,10\}$ and $A = B \neq C$ (iv) 343 (S) $P\!\left(\dfrac{E}{E_2}\right) = 0$[Note: $S \supseteq T$ denotes $S$ is a superset of $T$, means $S$ contains at least all elements of $T$.]Which of the following options is the only correct combination?

If $P(A) = 0.8$, $P(B) = 0.5$, and $P(B/A) = 0.4$, find $P(A \cap B)$.

The probability that the birthdays of six different persons will fall in exactly two calendar months

Class XII has sections A (40%) and B (60%). 20% students get into IIT. $P$(IIT from A)$=5P$(IIT from B). A student not selected in IIT is chosen. Probability he is from section B is

Neha lists all positive divisors of $(2010)^2$. She randomly selects 2 distinct divisors. Probability that exactly one is a perfect square is

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problems is:

An electrical system has open-closed switches $S_1$, $S_2$ and $S_3$ as shown. The switches operate independently of one another and the current will flow from $A$ to $B$ either if $S_1$ is closed or if both $S_2$ and $S_3$ are closed. If $P(S_1) = P(S_2) = P(S_3) = \frac{1}{2}$, then find the probability that the circuit will work.

Number of correct statements is $k$. Then $3k$ is: I) For two events $A,B$: $P(A\cap B)\ge P(A)+P(B)-1$. II) Number of symmetric relations on $\{1,2,3,4\}$ which are not reflexive is 20. III) $\lim_{n\to\infty}\{(a^{1/2}-a^{1/3})(a^{1/2}-a^{1/5})\cdots(a^{1/2}-a^{1/(2n+1)})\}=0$ if $a>1$.

The probability that in a random arrangement of the word MATHEMATICS, the two M's are not together given that the two A's are not together is

Four numbers are chosen at random (without replacement) from the set $\{1, 2, 3, \ldots, 20\}$.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 from an AP, then the set of all possible values of common difference is $\{\pm1, \pm2, \pm3, \pm4, \pm5\}$.

Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by both 2 and 3 is

In a game a coin is tossed $2n + m$ times and a player wins if he does not get any two consecutive outcomes same for at least $2n$ times in a row. The probability that player wins the game is

Find the probability that the 3 N's come consecutively in the arrangement of the letters of the word "CONSTANTINOPLE".

A six-faced dice is so biased that it is twice as likely to show an even number as an odd number when thrown. It is thrown twice, the probability that the sum of two numbers thrown is even is

Let $S = \{a, b, c, d, e, f, g\}$ and $x \in S$. A non-empty subset $A$ of $S$ is chosen at random. What is the probability that $x \in A$?

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:

A bag contains 4 white and 3 black balls. Another bag contains 3 white and 7 black balls. One ball is drawn from each bag. The probability that both are same colour is

Three coins are tossed simultaneously. The probability that all coins have the same face up, given that at least one head shows, is

In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question with four choices, only one answer being correct. The probability that he makes a guess is $\dfrac{1}{3}$ and the probability that he copies the answer is $\dfrac{1}{6}$. The probability that his answer is correct, given that he copies it, is $\dfrac{1}{8}$. Find the probability that he knew the answer to the question, given that he correctly answers.

If two different numbers are taken from the set $\{0, 1, 2, 3, \ldots, 10\}$; then, the probability that their sum as well as absolute difference are both multiple of 4, is

Lot $A$ consists of 5 good and 3 defective articles. Lot $B$ consists of 3 good and 5 defective articles. A new lot $C$ is formed by taking 3 articles from $A$ and 4 articles from $B$. The probability that an article chosen at random from $C$ is defective, is:

Given P(A ∪ B) = P(A ∩ B), then which of the following is correct?(1) P(A) + P(B) = 2P(A ∩ B)(2) P(A) = P(B)(3) A and B are equally likely(4) All of the above

Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively for the three critics. Find the probability that majority are in favour of the book.

One ticket is selected at random from 50 tickets numbered 00, 01, 02, … 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

A coin is tossed (m + n) times, m > n. Show that the probability of getting (at least) m consecutive heads is (n + 2)/2m+1.

A dice is thrown six times, it being known that each time a different digit is shown. The probability that a sum of 12 will be obtained in the first three throws is

Two players toss 4 coins each. The probability that they both obtain the same number of heads is

A man parks his car among n cars standing in a row, his car not being parked at an end. On his return he finds that exactly m of the n cars are still there. What is the probability that both the cars parked on two sides of his car have left?

In a multiple-choice question, there are four alternative answers of which one or more answers are correct. A candidate gets marks if he ticks all the correct answers. The candidate, being ignorant about the answers, decides to tick at random. How many attempts at least should he be allowed so that the probability of his getting marks in the question may exceed $\dfrac{1}{4}$?

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 colour is

A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are legible. The probability that it came from LONDON is

Since 7 coupons are numbered from 1 to 9 so that '9' is selected at least once, the required probability is:

Probability Questions (959)