Probability practice · Mixed difficulty

Two non-negative integers $x$ and $y$ are chosen at random with replacement. The probability that $x^2+y^2$ is divisible by 10, is

A dice is rolled three times. The probability of getting a larger number than the previous number each time is

Four persons are selected from a group of 4 men, 2 women and 3 children. The probability that exactly two of them are men is

Girl students constitute $10 \%$ of I year and $5 \%$ of II year at Roorkee University. During summer holidays $70 \%$ of the I year and $30 \%$ of II year students are given a project. The girls take turns on duty in canteen. The chance that I year girl student is on duty in a randomly selected day is

If $P\left( A \right) = P\left( B \right) = x$ and $P\left( {A \cap B} \right) = P\left( {A' \cap B'} \right) = \frac{1}{3}$, then $x$ is equal to

A five digit 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

Four persons are chosen at random from a group of 3 men, 2 women and 4 children. The chance that exactly 2 of them are children, is

Let $A$ and $B$ be independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then,

In a college 25% boys and 10% girls offer Mathematics. There are 60% girls in the college. If a Mathematics student is chosen at random, then the probability that the student is a girl, will be

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

Consider 5 independent Bernulli's trials each with probability of success $p$. If the probability of at least one failure is greater than or equal to $\frac{31}{32}$, then $p$ lies in the interval:

If $(1+3 p) / 3,(1-p) / 4$ and $(1-2 p) / 2$ are the probabilities of three mutually exclusive events, then the set of all values of $p$ is

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

In a bag there are 20 cards 10 names $A$ and another 10 names B. Cards are drawn randomly one by one with replacement, the probability that second $A$ comes before third $B$ is

A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then probability that both are dead is

Twenty five coins are tossed simultaneously. The probability that the fifth coin will fall with head upward, is

If $P\left( {A \cap B} \right) = \frac{1}{3},\;P\left( {A \cup B} \right) = \frac{5}{6},\;$ and $P\left( A \right) = \frac{1}{2}$ then which one of the following is correct?

Ram and Shyam throw with one dice for a prize of Rs. 88 which is to be won by the player who throws 1 first. If Ram starts, then mathematical expectation for Shyam is

The probability that Krishna will be alive 10 year hence is $\frac{7}{15}$ and Hari will be alive is $\frac{7}{10}$. The probability that both Krishna and Hari will be dead 10 years hence is

Suppose $A$ is a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are chosen at random. If the probability that $P$ is a subset of $Q$ is $243 / 1024$, then $n$ is equal to

A bag contains a mixed lot of red and blue balls. If two balls are drawn at random, the probability of drawing two red balls is five times the probability of drawing two blue balls and the probability of drawing one ball of each colour is six times the probability of drawing two blue balls. Then, the total number of red and blue balls in the bag is

If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 is

A and B are two independent events. The probability that both $\mathrm{A}$ and $\mathrm{B}$ occur is $1 / 6$ and the probability that at least one of them occurs is $2 / 3$. Find $8 \mathrm{P}(\mathrm{A})+9 \mathrm{P}(\mathrm{B})$ if $\mathrm{P}(\mathrm{A})>$ $\mathrm{P}(\mathrm{B})$.

Sum of two positive integers is 66 and $\mu$ is the maximum value of their product $S=\left\{x \in \mathbb{Z}, x(66-x) \geq \frac{5 \mu}{9}\right\}, x \neq 0$, then probability of $A$ when $A=\{x \in S ; x=3 k, x \in \mathbb{N}\}$ is:

A fair coin is tossed 100 times. The probability of getting tails an odd number of times, is

Three faces of a fair dice are yellow, two faces red and one blue. The dice is tossed three times. The probability that the colours, yellow, red and blue appear in the first, second and third tosses respectively is

Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has atleast one girl is

Three identical dice are rolled. The probability that same number appears on them, is

If $A$ and $B$ are two subsets of set $X=\{1,2,3,4,5\}$, then the probability that $A \cap B$ contains just two elements is

Let Ajay will not appear in JEE exam with probability $p=\frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q=\frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is