Probability practice · Mixed difficulty

Let $S=\left\{M=\left[a_{i j}\right], a_{i j} \in\right\}\{0,1,2\},\{1 \leq i, j \leq 2\}$ be a sample space and $A\{M \in S: M$ is invertible $\}$ be an even. Then $P(A)$ is equal to

If $P\left( B \right) = \frac{3}{4},\;P\left( {A \cap B \cap \bar C} \right) = \frac{1}{3}\;$ and $P\left( {\bar A \cap B \cap \bar C} \right) = \frac{1}{3},\;$ then $P\left( {B \cap C} \right)$ is

If $n$ positive integers are taken at random and multiplied together, the probability that the last digit of the product is 2, 4, 6 or 8, is

If $A$ and $B$ are two independent events, then the probability that only one of $A$ and $B$ occur is

In a bag there are 10 balls of either Black or White. Let $A_k$ be the event which contains exactly $k$ white balls and its probability is proportional to $k^2$ Now an another ball is drawn randomly. If $P(E)$ be the probability that the ball drawn is white and $P(E)=\frac{a}{b}$ Where $\operatorname{HCF}(a, b)=1$ then the value of $|\mathrm{a}-\mathrm{b}|$ equals

A person draws out two balls successively from a bag containing 6 red and 4 white balls. The probability that at least one of them will be red, is

The number of times a die must be tossed to obtain a 6 at least once with probability exceeding 0.9 is at least

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct.

Let $X$ be a binomially distributed random variable with mean 4 and variance $\frac{4}{3}$. Then, $54 P(X \leq 2)$ is equal to

Which of the following is true?

I. If the empty set $\phi$ and the sample space describe events, then $\phi$ is an impossible event.

II. In the above statement, the whole sample space ${S}$ is called the sure event.

An integer is chosen at random from first two hundred numbers. Then, the probability that the integer chosen is divisible by 6 or 8 is

From a pack of 52 cards, two cards are drawn one by one without replacement. The probability that first drawn card is a king and second is a queen, is

The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If $P\left( {A \cup B} \right) = 0.8,\,\,\,P\left( {A \cap C} \right) = 0.3,\,\,\,\,P\left( {A \cap B \cap C} \right) = 0.2,\,\,\,\,P\left( {B \cap C} \right) = \beta$ and $P\left( {A \cup B \cup C} \right) = \alpha ,$ where $0.85\, \le \alpha \le 0.95,$ then $\beta$ lies in the interval

Let $E^{\prime}$ denote the complement of an event $E$. Let $E, F$, $G$ be pairwise independent events such that $P(G)>0$ and $P(E \cap F \cap G)=0$. Then $P\left(E^{\prime} \cap F^{\prime} / G\right)$ equals

The probability that a number selected at random from the set of numbers 1, 2, 3, 4, …, 100 is a cube, is

If $\mathrm{A}$ and $\mathrm{B}$ are two events such that $P(A)=\frac{4}{7}, P(A \cap B)=\frac{3}{28}$ and the conditional probability $P\left(\frac{A}{A^c \cup B^c}\right)$ (where $A^c$ denotes the compliment of the event $\mathrm{A}$ ) is equal to $\lambda$, then the value of $\frac{13}{\lambda}$ is equal to

A bag contains 5 white, 6 green, 7 blue, 8 red and 9 black balls. If 3 balls are drawn one by one with replacement after each drawing, then the probability that $3^{\text {rd}}$ ball drawn is green is, if it is known that second ball is green:

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. One fruit is picked out from each basket. The probability that the fruits are both apples or both oranges, is

Let $A$ be a set containing $n$ elements. A subset $P$ of the set $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P$, and another subset $Q$ of $A$ is chosen at random. The probability that $P \cap Q$ contains exactly $m(m < n)$ elements, is

If a bag contains 50 tickets, numbered $1,2,3, \ldots, 50$ of which five are drawn at random and arranged in ascending order of magnitude $\left( {{x_1} < {x_2} < {x_3} < {x_4} < {x_5}} \right)$. The probability that $x_3=30$, is

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is

The probability that a leap year selected at random will contain either 53 Thursdays or 53 Fridays, is

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 replaced another inspector does the same thing and then so does a third inspector. The probability that atleast one of the inspectors finds a defective transistor, is equal to

If $A$ and $B$ are two events, the probability that exactly one of them occurs is not given by

Three dice are thrown. The probability of getting a sum which is a perfect square, is

A police-man fires six bullets on a dacoit. The Probability that the dacoit will be killed by one bullet is 0.6 . The Probability that dacoit is still alive is

Four fair dice $D_1, D_2, D_3$ atd $D_4$, each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously. The probability that $D_4$ shows a number on one of $D_1, D_2$ and $D_3$ is

The probability that the 13th day of a randomly chosen month is a Friday, is

Let $A$ and $B$ be two events such that $P(B \mid A)=\frac{2}{5}, P(A \mid B)=\frac{1}{7}$ and $P(A \cap B)=\frac{1}{9}$, Consider $(S 1) P\left(A^{\prime} \cup B\right)=\frac{5}{6},(S 2) P\left(A^{\prime} \cap B^{\prime}\right)=\frac{1}{18}$. Then