Probability practice · Mixed difficulty

If two events $A$ and $B$ such that $P\left(A^{\prime}\right)=0.3, P(B)=0.5$ and $P(A \cap B)=0.3$, then $P\left(B \mid A \cup \mathrm{B}^{\prime}\right)$ is

Six faces of a die are marked with the numbers $1,-1,0,-2,2$ and 3 . The die is thrown thrice. The probability that the sum of the numbers thrown is six, 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 IV. The probabilities of the student passing in tests I, II, III are $\mathrm{p}, \mathrm{q}$ and $\frac{1}{2}$ respectively. The probability that the student is successful is $\frac{1}{2}$ then the relation between $\mathrm{p}$ and $q$ is given by

A coin is tossed $2n$ times. The chance that the number of times one gets head is not equal to the number of times one gets tail, is

The probability that a man will live 10 more years is 1/4 and the probability that his wife will live 10 more years is 1/3. Then the probability that neither will be alive in 10 years, is

Five coins whose faces are marked 2, 3 are tossed. The chance of obtaining a total of 12 is

A box contains 100 tickets numbered 1, 2, …, 100. Two tickets are choosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. The probability that the minimum number is 5 is

A ship is fitted with three engines $E_1, E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{4}$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that ship is operational and let $X_1, X_2$ and $X_3$ denote respectively the events that the engines $E_1, E_2$ and $E_3$ are functioning. Which of the following is not true?

A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the same colour, is

A man is known to speak truth 3 out of 4
times. He takes out a card at random from a well shuffled
pack of 52 playing cards, and reports it is a king. The
probability that its actually a king is

A book contains 1000 pages. A page is chosen at random. The probability that the sum of the digits of the marked number on the page is equal to 9 , is

There are 4 envelopes corresponding to 4 letters. If the letters are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes, is

If $\overline{{E}}$ and $\overline{{F}}$ are the complementary events of events ${E}$ and ${F}$ respectively and if $0 < {P}({F}) < 1$, then

A bag contain 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomly, the probability that it is a black or red ball, is

Six faces of an unbiased die are numbered with 2, 3, 5, 7, 11 and 13. If two such dice are thrown, then the probability that the sum on the uppermost faces of the dice is an odd number, is

A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. The probability that it is actually a six, is

If events are independent and $P\left( A \right) = \frac{1}{3},\;P\left( B \right) = \frac{1}{3},\;P\left( C \right) = \frac{1}{4},\;$ then $\mathrm{P}\left( {A' \cap B' \cap C'} \right)$ is equal to

One Indian and four American men and their wives are to be seated randomly around a circular table. The conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

If two squares are chosen at random on a chess board, the probability that they have a side in common is

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

Consider two events $A$ and $B$ such that $P\left( A \right) = \frac{1}{4},\;P\left( {\frac{B}{A}} \right) = \frac{1}{2},\;P\left( {\frac{A}{B}} \right) = \frac{1}{4}.$ For each of the following statements, which is true?
1. $P\left( {{A^c} + {B^c}} \right) = \frac{3}{4}\;$
2. The events $A$ and $B$ are mutually exclusive.
3. $P(A/B)\, + \,P(A/{B^c})\, = \,1$

$A$ and $B$ toss a coin alternatively till one of them gets a head and wins the game. If $A$ begins the game, the probability $B$ wins the game is

If the probability for $A$ to fail in an examination is 0.2 and that for $B$ is 0.3, then the probability that either $A$ or $B$ fails, is

If $A$ and $B$ are two events such that $P\left( A \right) \ne 0$ and $P\left( B \right) \ne 1,$ then $P\overline {\left( {\frac{A}{B}} \right)}$ is equal to

Two numbers a and b are chosen at random from the set of first 30 natural numbers. The probability that ${a^2} - {b^2}$ is divisible by 3 is

The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are chosen from the set
$\{1,2,3,4,5,6,7,8\}.$ The probability of this equation having repeated roots is:

The records of a hospital show that 10% of the cases of a certain disease are fatal. If 6 patients are suffering from the disease, then the probability that only three will die, is

Two dice are thrown simultaneously. What is the probability of obtaining a multiple of 2 on one of them and a multiple of 3 on the other

Let ${p}$ denote the probability that a man aged $x$ years will die in a year. The probability that out of ${n}$ men ${A}_1,{A}_2, {A}_3, \ldots \ldots {A}_n$ each aged $x, {A}_1$ will die in a year and will be the first to die is

A fair coin is tossed a fixed number of times. If the probability of getting 4 heads equals the probability of getting 7 heads, then the probability of getting 2 heads is