Probability practice · Medium

Two small squares on a chess board are chosen at random. Then, the probability that they have a common side, is

If the numbers $1,2,3,4,5,6$ are arranged in a circle, then the probability that the sum of any two consecutive numbers must not be equal to 7 is

Two distinct numbers are chosen at random from the first 15 natural numbers. The probability that the sum will be divisible by 3 is

A box contains 9 tickets numbered 1 to 9 inclusive. If 3 tickets are drawn from the box without replacement. The probability that they are alternatively either {odd, even, odd} of {even, odd, even} is

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 returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is

Let $A, B$ and $C$ be three events such that $P(C)=0$.

Statement-1: $P(A \cap B \cap C)=0$

Statement-2: $P(A \cup B \cup C)=P(A \cup B)$

Two numbers are randomly selected from
the first 100 natural numbers. The probability that the
product of the numbers is divisible by 7 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

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 $P\left\{\frac{1}{2} < X < \frac{5}{2}|X>1|\right\}=$

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

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

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

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:

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