Probability Questions with Solutions

If $0<P\left(A\right)<1$, $0<P\left(B\right)<1$ and $P\left(A\cap B\right)=P\left(A\right)P\left(B\right)$, then

A four-digit number is formed using the digits 1, 2, 3, 4, 5 without repetition. The probability that it is divisible by 3 is

There are two circles in the xy-plane whose equations are $x^2+y^2-2y=0$ and $x^2+y^2-2y-3=0$. A point $\left(x,y\right)$ is chosen at random inside the larger circle. Then the probability that the point has been taken from the smaller circle is

A chain of video stores sells three different brands of DVD players. Of its DVD player sales, 50% are brand 1, 30% are brand 2 and 20% are brand 3. Each manufacturer offers one year warranty on parts and labor. It is known that 25% of brand 1 DVD players require warranty repair work whereas the corresponding percentage for brands 2 and 3 are 20% and 10% respectively. The probability that a randomly selected purchaser has a DVD player that will need repair while under warranty, is:

A box contains 3 coins, one coin is fair, one coin is two headed and one coin is weighted, so that the probability of heads appearing is $\frac{1}{3}$. A coin is selected at random and tossed, then the probability that head appears is

If a fair die is rolled successively, then the probability that 1 appears for the first time in an even-numbered throw is

Out of $2n+1$ tickets, which are consecutively numbered, three are drawn at random. Then the probability that the numbers on them are in arithmetic progression is

A is targeting B, B and C are targeting A. Probabilities of hitting the target by A, B and C are $\frac{2}{3}$, $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If A is hit, then the probability that B hits the target and C does not is

In an entrance test there are multiple choice questions, with 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 the student gets the correct answer to a question, then the probability that he was guessing is

A man is known to speak the truth 2 out of 3 times. He throws a die with 1 to 6 on its faces and reports that it is 1. Then the probability that it is actually 1 is

If $E_1$ and $E_2$ are two events associated with a random experiment such that $P\left(E_2\right)=0.35$, $P\left(E_{1}or{E}_2\right)=0.85$ and $P\left(E_{1}and{E}_2\right)=0.15$, then $P\left(E_1\right)$ is

Two persons A and B agree to meet on 20 April 2018 between 6 pm and 7 pm, with the understanding that each will wait no longer than 20 minutes for the other. What is the probability that they meet?

Three numbers a, b and c are chosen at random (without replacement) from among the numbers 1, 2, 3, ..., 99. The probability that $a^3+b^3+c^3-3abc$ is divisible by 3 is

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

A committee of 5 is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is

A speaks the truth in 40% and B in 50% of the cases. The probability that they contradict each other while narrating some incident is

Let A and B be two events defined on a sample space $\Omega$. Suppose $A^C$ denotes the complement of A relative to the sample space $\Omega$. Then the probability $P\left(\left(A\cap B^C\right)\cup \left(A^C\cap B\right)\right)$ equals

Region R is defined as the region in the first quadrant satisfying the condition $x^2+y^2<4$. Given that a point $P=\left(r,s\right)$ lies in R, what is the probability that $r>s$?

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

The captains of five cricket teams, including India and Australia, are lined up randomly next to one another for a group photo. What is the probability that the captains of India and Australia will stand next to each other?