Probability Questions with Solutions

In a tuition batch of two students, the probability that $X$ will pass the exam is $\frac{2}{5}$ and that of $Y$ is $\frac{3}{4}$. What is the probability that neither of $X$ and $Y$ will pass the exam? Assume that the outcomes of exams for $X$ and $Y$ are independent of each other.

Let $A_1,A_2,A_3$ be three events in a sample space with the condition $A_1\cap A_2\ne \varphi$. Then always:

A die is rolled twice independently. What is the probability that either the first die shows a number no less than 4 or the second die shows at least 4?

A bookshelf has 5 red books, 3 blue books, and 4 green books. If you randomly select a book from the shelf, what is the probability that it is blue?

A bag contains 5 yellow, 4 black and 3 green balls. If 3 balls are drawn at random, find the probability that no black ball is chosen.

A speaks the truth in 60% of cases and B in 50% of cases. In what fraction of cases are they likely to contradict each other while narrating the same incident?

Atal speaks the truth in 70% and George speaks the truth in 60% of cases. In what percentage of cases are they likely to contradict each other in stating the same fact?

If A and B are two events such that $P\left(A\cup B\right)=\frac{5}{6}$, $P\left(A\cap B\right)=\frac{1}{3}$ and $P\left(\overline{B}\right)=\frac{1}{2}$, then the events A and B are

Two fair dice are tossed. What is the probability that the total score is a prime number?

Let $P\left(E\right)$ denote the probability of event $E$. Given $P\left(A\right)=1$ and $P\left(B\right)=\frac{1}{2}$, the values of $P\left(A|B\right)$ and $P\left(B|A\right)$ respectively are

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shots are $0.4$, $0.3$, $0.2$ and $0.1$ respectively. The probability that the gun hits the plane is

A coin is tossed three times. The probability of getting heads and tails alternately is

A man has 5 coins: 2 are double-headed, 1 is double-tailed and 2 are normal. He picks a coin at random and tosses it. The probability that the lower face is a head is:

An anti-aircraft gun can fire a maximum of four shots at an enemy plane. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits the plane is:

A six-faced die is a biased one. It is thrice more likely to show an odd number than to show an even number. It is thrown twice. The probability that the sum of the numbers in the two throws is even is

A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is

The probabilities of occurrence of two events E and F are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither E nor F occurs 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 $E_1$ 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