Probability practice · Mixed difficulty

Let $n(A-B)=25+x, n(B-A)=2 x$ and $n(A \cap B)=2 x$. If $n(A)=2 n(B)$ then ' $x$ ' is

7 white balls and 3 black balls are placed in a row at random. The probability that no two black balls are adjacent is

If $A$ and $B$ are two events such that $P(A)=\frac{2}{5}$ and $P(A \cap B)=\frac{3}{20}$, then the conditional probability $P\left(A / A^{\prime} \cup B^{\prime}\right)$, where $A^{\prime}$ denotes the complement of $A$, is equal to

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :

The probability that a candidate secures a seat in Engineering through “EAMCET” is $1/10.$ 7 candidates are selected at random from a centre. The probability that exactly two will get seats is

A sportsman's chance of shooting an animal at a distance $r > a$ ("a" is constant) is given to be $\frac{a^2}{r^2}$. He fires at $r=2 a$ and if he misses, then again fires at $r=3 a$. He repeats the same process at $r=4 a, 5 a$ and $6 a$. When he misses at $r=6 a$, the animal escapes into the jungle. If the odds against the sportsman are ${p}: {q}$, then ${q}-{p}$ is ____.

The probability that the 6th day of a randomly chosen month of a year is a Sunday, is

Six coins are tossed simultaneously. The probability of getting at least 4 heads is

For two events $A$ and $B,$ if $P\left( A \right) = P\left( {\frac{A}{B}} \right) = \frac{1}{4}\;\,\mathrm{and}\,\,P\left( {\frac{B}{A}} \right) = \frac{1}{2},$ then

A lottery sells $n^2$ tickets and declares $n$ prizes. If a man purchases $n$ tickets, the probability of his winning exactly one prize is

From a well-shuffled pack of 52 cards, four cards are selected at random. The probability of drawing exactly 2 spades and exactly 2 aces is

A lot consists of 102 good pencils, 6 with minor defects and 2 with major defects. A pencil is choosen at random. The probability that this pencil is not defective is

A five digits number is formed by writing the digits 1, 2, 3, 4, 5 in a random order without repetitions. Then the probability that the number is divisible by 4, is

If $a \in \left[ { - 20,0} \right],$ then the probability that the graph of the function $y = 16\;{x^2} + 8\left( {a + 5} \right)x - 7\;a - 5$ is strictly above the $x$-axis is

Let $E_1, E_2, E_3$ be three mutually exclusive events such that $P\left(E_1\right)=\frac{2+3 p}{6}, P\left(E_2\right)=\frac{2-p}{8}$ and $P\left(E_3\right)=\frac{1-p}{2}$. If the maximum and minimum values of $p$ are $p_1$ and $p_2$, then $\left(p_1+p_2\right)$ is equal to :

If the probability that a person is not a swimmer is 0.3 , then the probability that out of 5 persons 4 are swimmers is

Nine identical balls are numbers 1, 2, ...9. are put in a bag. A draws a ball and gets the number $a$. The ball is put back the bag. Next $B$ draws a ball gets the number $b$. The probability that $a$ and $b$ satisfics the inequality $a-2 b+10 > 0$ is

The probability that a company executive will travel by train is $\frac{2}{3}$ and that he will travel by plane is $\frac{1}{5}.$ The probability of his journey by train or plane is

If a dice is thrown twice, the probability of occurrence of 4 at least once, is

You are given a box with 20 cards in it. 10 of these cards have letter $I$ printed on them. The other ten have the letter ${T}$ printed on them. If you pick up 3 cards at random and keep them in same order, the probability of making the word $IIT$, is

If $A$ and $B$ are arbitrary events, then

If a dice is thrown twice, the probability of occurrence of 4 at least once is

If $A$ and $B$ are two independent events such that $P(\bar{A} \cap B)=2 / 15$ and $P(A \cap \bar{B})=1 / 6$, then $P(B)$, is

If $A$ and $B$ are two events such that $P\left( {A \cup B} \right) = \frac{5}{6},$ $P\left( {A \cap B} \right) = \frac{1}{3}$ and $P\left( {\bar B} \right) = \frac{1}{3},$ then the value of $P\left( A \right)$ is

If the random variable $X$ takes the values ${x_1},\;\,{x_2},\,{x_{10}}$ with probability $P\left( {X = {x_1}} \right) = ki,$ then the value of $k$ is equal to

The probability that a leap year will have 53 Fridays or 53 Saturdays, is

For a party 8 guests are invited by a husband and his wife. They sit for a dinner around a round table. The probability that the husband and his wife sit together, is

An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses needed, is

If $\left( {\;\frac{{1 + a}}{3}} \right)\;\,\mathrm{and}\,\;\left( {\;\frac{{1 - a}}{4}} \right)$ are probability of two mutually exclusive events, then set of all values of $a$ is

A bag contains 3 same balls and 3 different balls of three different colours. Two balls are drawn randomly with replacement. The probability they have same colour is $m$. Again four balls are drawn one by one with replacement, then probability of getting three same balls is $n$. The value of $m$. $n$ is