Probability practice · Mixed difficulty

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are refurned to the bag. If now a ball is drawn at random from the bog, then the probability this ball drawn is red, is

The probability that the three cards drawn from a pack of 52 cards, are all black, is

If $P(A \cap B)=\frac{1}{2}, P(\bar{A} \cap \bar{B})=\frac{1}{3}, P(A)=p, P(B)=2 p$, then the value of ${p}$ is given by

The odds that $A$ speaks the truth are $3: 2$ and the odds that $B$ speaks the truth are $5: 3$. The percentage of cases they are likely to contradict each other on an identical point is

In a company, there are seven machines and the probability of any of them to be out of service on a day is $\frac{1}{3}.$ If the probability that atmost 3 machines will be out of service on the same day is $\frac{16 k}{3^7},$ then $k$ is equal to

The probability distribution of a random variable is $P(X)=\left\{\begin{array}{c}\frac{X}{15} ; X=1,2,3,4,5 \\ 0, \text { otherwise }\end{array}\right.$. Then $P\left\{\frac{1}{2} < X < \frac{5}{2}|X>1|\right\}=$

The mean and variance of a binomial variable $X$ are 2 and 1 respectively. The probability that $X$ takes values greater than 1, is

A biased coin with probability $p, 0 < p < 1$, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $2 / 5$, then p equals

Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$, then the probability of both of them occur together is

Two coins are available, one fair and the other two headed. Choose a coin and toss if once assume that the unbiased coin is chosen with probability $\frac{3}{4}$. Given that the outcome is head, the probability that the two-headed coin was chosen is

An insurance salesman sells policies to 5 men, all of identical age and in good health. The probability that a man of this particular age will be alive after 30 years is 2/3. The probability that after the lapse of 30 years all the five persons will be alive, is

In an entrance examination there are multiple choice questions. There are four possible answers to each question of which one is correct. The probability that a student knows the answer to a question is $90 \%$. If he gets the correct answer to the question, then the probability that he was guessing is

Each of $a$ and $b$ can take values 1 or 2 with equal probability. The probability that the equation $a x^2+b x+1=0$ has real roots, is equal to

Let $\mathrm{\omega }$ be a complex cube root of unity with $\mathrm{\omega \;} \ne 1$. A fair die is thrown three times. If ${r_1},\;{r_2}\;$ and ${r_3}$ are the numbers obtained on the die, then the probability that ${\mathrm{\omega }^{{r_1}}} + {\mathrm{\omega }^{{r_2}}} + {\mathrm{\omega }^{{r_3}}} = 0$ is

If the sides of a triangle are decided by the throw of a single dice thrice, the probability that triangle is of maximum area given that it is an isosceles triangle, is

A man alternately tosses a coin and throws a dice beginning with the coin. The probability that he gets a head before he gets 5 or 6 in the dice is

A game is played with fair cubic die which has one red side, two blue sides and three green sides. The result is the color of the top side after the dice is rolled. If the die is rolled repeatedly then the probability that second blue result occurs on or before the tenth roll can be expressed in the form of $\frac{3^p-2^q}{3^r}$ where $p, q, r$ are positive integers then the value of $p+r-q$ is

Let $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ be three events such that the probability that exactly one of $\mathrm{A}$ and $\mathrm{B}$ occurs is $(1-k)$, the probability that exactly one of $\mathrm{B}$ and $\mathrm{C}$ occurs is $(1-2 k)$, the probability that exactly one of $\mathrm{C}$ and $\mathrm{A}$ occurs is $(1-k)$ and the probability of all $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ occur simultaneously is $k^2$, where $0 < k < 1$. Then the probability that at least one of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ occur is:

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, the probability that not all letters are placed in correct envelopes is

Given that the sum of two non-negative quantities is 200, the probability that their product is not less than $3 / 4$ times their greatest product value is