Probability Questions (959)

Six different balls are put in three different boxes, no box being empty. The probability of putting balls in the boxes in equal numbers is :

The probability so that $r$ $1 \times 1$ squares which are selected from a $m \times n$ chess board such that no two of them share the same row or same column is :

A business man is expecting two telephone calls. Mr Walia may call any time between 2 p.m and 4 p.m. while Mr Sharma is equally likely to call any time between 2.30 p.m. and 3.15 p.m. The probability that Mr Walia calls before Mr Sharma is :

A machine containing $n$ different balls, when switched on, can throw up any number of balls one by one. The probability of throwing $r$ balls is directly proportional to $r$. Given that a particular ball is the first ball to pop up, the probability that machine has thrown up all the balls is:

From $4m+1$ tickets numbered as $1, 2, \ldots 4m+1$. Three tickets are chosen at random. The probability that the numbers are in A.P. with even common difference is

Let $A$ and $B$ be two events such that $P\left(A \cap B^{c}\right) = 0.20, P\left(A^{c} \cap B\right) = 0.15, P\left(A^{c} \cap B^{c}\right) = 0.1$, then $p(A/B)$ is equal to,

A point is selected at random inside a circle. The probability that the point is closer to the centre of the circle than to its circumference :

A point is selected at random inside an equilateral triangle whose side length is 3. The probability its distance to any corner is greater than 1 is

A point $X$ is selected at random from a line segment $AB$ with mid point $O$. The probability that the line segments $AX, XB$ and $AO$ can form a triangle is :

Players $P_1, P_2, P_3, \ldots, P_n$ of equal skill, play a game consecutively in pairs as $P_1P_2, P_2P_3, P_3P_4, \ldots, P_nP_1, \ldots$ and any player who wins two consecutive games (i.e $k$ and $(k+1)$th game) wins the match. If the chance that the match is won at the $r$th game is $k$ then:

Two persons $A$ and $B$ have respectively $n+1$ and $n$ coins, which they toss simultaneously. Then probability $P$ that $A$ will have more heads then $B$ belongs:

If $A$ and $B$ are two events such that $P(A) = \frac{3}{4}$ and $P(B) = \frac{5}{8}$ then:

A student has to match historical events viz., Dandi march, Quit India Movement and Mahatma Gandhi's assassination with the years 1948, 1930 and 1942. The student has no knowledge of the correct answer decides to match the events and years randomly. Let $E_i(0 \leq i \leq 3)$ denote the event that the student gets exactly $i$ correct answers, then which of the following is/are NOT correct?

A bag contains 20 blue marbles, 12 red marbles and some other number of green marbles. If the probability of drawing green marble in one try is $\frac{1}{y}$ then which of the following statements is/are correct?

A dice is rolled three times. Let $E_1$ denote the event of getting a number larger than the previous number each time and $E_2$ denote the event that the numbers (in order) form an increasing $AP$ then:

One card is missing from a pack of cards. Let $A$ be the event that missing card is a spade. Then two cards are drawn, and $S$ be the event that they are spades then:

A square is inscribed in a circle. If $p_1$ is the probability that a randomly chosen point of the circle lies within the square and $p_2$ is the probability that the point lies outside the square, then:

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

The letters of the word PROBABILITY are written down at random in a row. Let $E_1$ denote the event that two $i$, $s$ are together and $E_2$ denote the event that two $B$'s are together, then:

$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occurs is $1/6$ and the probability that neither of them occurs is $1/3$. Then the probability of the occurrence of $A$ may be:

The probabilities of events $A \cap B$, $A$, $B$ and $A \cup B$ are respectively in A.P. with second term equal to the common difference. Therefore, $A$ and $B$ are:

5 players of equal strength play one each with each other. $P(A) =$ probability that at least one player wins all matches he (they) plays. $P(B) =$ probability that at least one player loses all his (their) matches. Then:

A family has three children. Event 'A' is that the family has at most one boy. Event 'B' is that family has at least one boy and one girl. Event 'C' is that the family has at most one girl. Then:

A certain coin is tossed with probability of showing head being 'p'. Let 'q' denote the probability that when the coin is tossed four times the number of heads obtained is even. Then:

A bag contains four tickets marked with numbers 112, 121, 211, and 222. One ticket is drawn at random from the bag. Let $E_i$ (i = 1, 2, 3) denote the event that $i^{th}$ digit on the ticket is 2. Then:

Sudheer has 3 identical white cubes and wants to turn them into dice. For each cube he makes two independent decisions: 1. Colour: Red or Blue (each with probability $\frac{1}{2}$). 2. Pattern: Dots or Digits (each with probability $\frac{1}{2}$). He places all 3 dice in a bag. Renuka randomly selects one die and notes its colour and pattern. Define: - Event $A$: The selected die is Red. - Event $B$: The selected die is Dotted. The probability that events $A$ and $B$ are independent is:

A person has 6 cards: A, K, 2, Q, 10, 9. The person randomly draws the cards one by one with replacement till he gets 3 consecutive A's. If $P_n$ represents the probability that at least $n$ cards are drawn, then:

A person has 6 cards: A, K, 2, Q, 10, 9. The person randomly draws the cards one by one with replacement till he gets 3 consecutive A's. If $P_n$ represents the probability that at least $n$ cards are drawn, then:

Students $S_1,S_2,S_3$ agree: $P(E_1)=\frac{1}{2}$ (no one scores $>75\%$), $P(E_2)=\frac{1}{5}$ ($S_3$ scores $>75\%$), $P(E_3)=\frac{1}{10}$ ($S_3$ scores $>75\%$ given at least two score $>75\%$), $P(E_4)=\frac{1}{4}$ (exactly one scores $>75\%$). The probability that only $S_3$ scores above 75\% marks is:

A student answers all true-false questions. He knows some answers and guesses the rest. $P(\text{correct}|\text{guessed})=\frac{1}{2}$. Given the student's answer is correct, the probability it was guessed is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question, given that his answer is correct, is:

A student answers all true-false questions. He knows some answers and guesses the rest. $P(\text{correct}|\text{guessed})=\frac{1}{2}$. Given the student's answer is correct, the probability it was guessed is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question, given that his answer is correct, is:

Students $S_1,S_2,S_3$ agree: $P(E_1)=\frac{1}{2}$ (no one scores $>75\%$), $P(E_2)=\frac{1}{5}$ ($S_3$ scores $>75\%$), $P(E_3)=\frac{1}{10}$ ($S_3$ scores $>75\%$ given at least two score $>75\%$), $P(E_4)=\frac{1}{4}$ (exactly one scores $>75\%$). The probability that only $S_3$ scores above 75\% marks is:

Let $S$ be the set of all possible values of the determinant of a matrix $M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d$ are chosen independently and uniformly from $\{0,1\}$. Let $X=\det(M)$. Consider the quadratic $t^2-\gamma t+\delta=0$, where $\gamma$ and $\delta$ are chosen independently from $S$ with probabilities proportional to their frequency in $M$. The probability that the roots are non-real complex is:

Let $S$ be the set of all possible values of the determinant of a matrix $M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d$ are chosen independently and uniformly from $\{0,1\}$. Let $X=\det(M)$. Consider the quadratic $t^2-\gamma t+\delta=0$, where $\gamma$ and $\delta$ are chosen independently from $S$ with probabilities proportional to their frequency in $M$. The probability that the roots are non-real complex is:

Sudheer has 3 identical white cubes and wants to turn them into dice. For each cube he makes two independent decisions: 1. Colour: Red or Blue (each with probability $\frac{1}{2}$). 2. Pattern: Dots or Digits (each with probability $\frac{1}{2}$). He places all 3 dice in a bag. Renuka randomly selects one die and notes its colour and pattern. Define: - Event $A$: The selected die is Red. - Event $B$: The selected die is Dotted. The probability that events $A$ and $B$ are independent is:

In Experiment I, two colored unbiased coins are tossed. Two subsets $P$ and $Q$ of 2 elements each are chosen randomly from sample space $S$ with replacement. In Experiment II, the same coins are tossed; the experiment is successful if the outcome matches any element of $P$ or $Q$. The probability that Experiment II is NOT successful is:

Machines A, B, C make 20%, 30%, 50% of bolts; defect rates 3%, 4%, 2%. Given a bolt is defective, probability it's from C is

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of k is ___.

NTA Test 13 (Numerical) The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0%, is

$2^n$ players of equal strength are playing a knock out tournament. If they are paired randomly in all rounds, the probability that out of two particular players $S_1$ and $S_2$ exactly one will reach in semi final is $(n \in N, n \geq 2)$:

A machine containing $n$ different balls, when switched on, can throw up any number of balls one by one. The probability of throwing $r$ balls is directly proportional to $r$. Given that a particular ball is the first ball to pop up, the probability that machine has thrown up all the balls is:

A student appears for test I, II and III. The student is successful if he passes either in test I, II or I, III. The probability of the student passing in test I, II and III are respectively $p$, $q$ and $1/2$. If the probability of the student to be successful is $1/2$ then :

A family has three children. Event 'A' is that the family has at most one boy. Event 'B' is that family has at least one boy and one girl. Event 'C' is that the family has at most one girl. Then:

A point is selected at random inside a circle. The probability that the point is closer to the centre of the circle than to its circumference :

Three smallest squares are chosen randomly on a chess board are the probability that these squares have exactly two corners, but no side common is:

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

$A$ and $B$ play a game of tennis. The situation of the game is as follows; if one scores two consecutive points after a deuce he wins; if loss of a point is followed by win of a point, it is deuce. The chance of a server to win a point is $2/3$. The game is at deuce and $A$ is serving. Probability that $A$ will win the match is : (Serves are changed after each game)

A family has three children. Event 'A' is that the family has at most one boy. Event 'B' is that family has at least one boy and one girl. Event 'C' is that the family has at most one girl. Then:

Entries of a $2 \times 2$ determinant are chosen from the set $\{-1, 1\}$. The probability that determinant has zero value is:

A dice is rolled three times. Let $E_1$ denote the event of getting a number larger than the previous number each time and $E_2$ denote the event that the numbers (in order) form an increasing $AP$ then: