Probability Questions with Solutions

The coefficients $a$, $b$, $c$ of $a{x}^2+bx+c=0$ are chosen by tossing 3 fair coins, where a head means 1 and a tail means 2. The probability that the roots are imaginary is

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

India plays 4 matches. In each match the probabilities of scoring $0$, $1$ and $2$ points are $0.45$, $0.05$ and $0.50$ respectively. The probability of scoring at least $7$ points is

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

A and B throw a die in succession to win a bet with A starting first. Whoever throws 1 first wins Rs. 110. What are the respective expectations of A and B?

A and B are independent witnesses. The probability that A speaks the truth is $x$ and the probability that B speaks the truth is $y$. If both agree on a statement, the probability that the statement is true is:

The probability that a man who is 85 years old will die before attaining the age of 90 is $\frac{1}{3}$. $A_1,A_2,A_3,A_4$ are four persons aged 85 years. The probability that $A_1$ will die before attaining 90 and will be the first to die 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:

If $A$ and $B$ are two events such that $P\left(A\prime \right)=0.3$, $P\left(B\right)=0.5$ and $P\left(A\cap B\right)=0.3$, then $P\left(B\mid A\cup B\prime \right)$ is

A pair of unbiased dice is rolled together till a sum of either $5$ or $7$ is obtained. The probability that $5$ comes before $7$ is

A letter is taken at random from the letters of the word STATISTICS and another letter is taken at random from the letters of the word ASSISTANT. The probability that they are the same letter is

A bag contains $6$ red and $4$ green balls. A fair die is rolled and a number of balls equal to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red 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 letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is

A problem in Mathematics is given to 3 students A, B and C. If the probability of A solving the problem is $\frac{1}{2}$ and B not solving it is $\frac{1}{4}$. The whole probability of the problem being solved is $\frac{63}{64}$, then what is the probability of solving it by C?

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

The probability that a man who is $x$ years old will die in a year is $p$. Then, amongst $n$ persons $A_1,A_2,\dots ,A_n$, each $x$ years old now, the probability that $A_1$ will die in one year and be the first to die is