Probability practice · Mixed difficulty

For three events $A, B$ and $C, P$ (exactly one of the events $A$ or $B$ occurs $)=P$ (exactly one of the events $B$ or $C$ occurs) $=P($ exactly one of the events $C$ or $A$ occurs $)=p$ and $P$ (all the three coents occur simultaneously) $= {p^2}$, where $0 < p < 1/2$. Then the probability of atleast one of the three events $A, B$ and $C$ occurring is

There are 7 seats in a row. Three persons take seats at random. The probability that the middle seat is always occupied and no two persons are consecutive is

What is the probability that for S's come consecutively in the woord 'MISSISSIPPI'?

At a telephone enquiry system the number of phone calls regarding relevant enquiry follow poisson distribution with an average of 5 phone calls during 10 min time intervals. The probability that there is at the most one phone cell during a 10 min time period, is

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of X is:

A bag $X$ contains 2 white and 3 black bails and another bag $Y$ contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, the probability for the ball chosen be white, is

Two fair dice are rolled once. Then the mean of sum of the numbers on them is

Probability that a student will succeed in I.I.T. entrance test is 0.2 and that he will succeed in Roorkee entrance test is 0.5. If the probability that he will successful at both the places is 0.3, then the probability that he does not succeed at both the places is

A die is rolled, let ${E}$ be the event "die shows 4" and ${F}$ be the event "die shows even number". Then

I. ${E}$ and ${F}$ are mutually exclusive.

II. ${E}$ and ${F}$ are not mutually exclusive.

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is

Three dice are thrown. The probability that the sum of the number appearing is 15, is

The number of ways in which a pack of 52 cards of four different suits can be distributed equally among four players so that each player gets the Ace, King, Queen and Knave of the same suit is

The probability that certain electronic conponent fails when first used in 0.10 . If it does not fail immediately, the probability that it lasts for one year is 0.99 . The probability that a new component will last for one year is

The probability that a man can hit a target is $3/4.$ He tries 5 times. The probability that he will hit the target at least three times 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 die is tossed thrice. If event of getting an even number is a success, then the probability of getting at least two successes is

A consignment of 15 record players contains 4 defectives. The record players are selected at random, one by one, and examined. The ones examined are not put back. The probability that 9th one examined is the last defective, is

A room has 3 lamps. From a collection of 10 light bulbs of which 6 are not good, a person selects 3 at random and puts them in a socket, the Probability that he will have light, is

Two cards are drawn from a well shuffled deck of 52 cards. The probability that one is red card and the other is a queen is

If three natural numbers from 1 to 100 are selected randomly, then probability that all are divisible by both 2 and 3, is

The mean and the variance of a binomial distribution are 4 and 2 respectively. Then, the probability of 2 successes is

If the papers of 4 students can be checked by any one of the 7 teachers, then the probability that all the 4 papers are checked by exactly 2 teachers is

If $A$ and $B$ are independent events of a random experiments such that $P\left( {A \cap B} \right) = \frac{1}{6}$ and $P\left( {\bar A \cap \bar B} \right) = \frac{1}{3},\;$ then $P\left( A \right)$ is equal to

In the above question the probability that the binary operation is non-commutative is

If $E$ and $F$ are events such that ${P}({E})=\frac{1}{4}, {P}({F})=\frac{1}{2}$ and $P(E$ and $F)=\frac{1}{8}$, then,
I. ${P}({E}\text{ or }{F})=\frac{5}{8}$
II. ${P}(\text{not }{E} \text{ and not } {F})=\frac{3}{8}$

If $E$ and $F$ are two independent events such that $0 < P\left( E \right) < 1$ and $0 < P\left( F \right) < 1,\;$ then

If $P(A)=\frac{1}{4}, P(\bar{B})=\frac{1}{2}$ and $P(A \cup B)=\frac{5}{9}$ then $P(A / B)$ is

If $P\left( A \right) = 1/3,P\left( B \right) = 1/2$ and $P\left( {A \cup B} \right) = 5/6,$ then events $A$ and $B$ are

A bag contains 6 white and 6 black socks. A man randomly takes out two socks from the bag. The probability that socks are the same colour is:

Six dice are thrown simultaneously. The probability that all of them show the same face, is