Probability Questions

The probability that $A$ speaks truth is $\frac{4}{5},$ while this probability for $B$ is $\frac{3}{4}.$ The probability that they contradict each other when asked to speak on a fact is

A random variable $X$ has the probability distribution:

For the events $E=\{X\text{ is a prime number}\}$ and $F=\{X<4\},$ the probability $P(E\cup F)$ is

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

At a telephone enquiry system, the number of phone calls regarding relevant enquiry follows a Poisson distribution with an average of $5$ phone calls during $10\text{-minute}$ time intervals. The probability that there is at most one phone call during a $10\text{-minute}$ time period is:

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly $9$ twice is:

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are $0.3$ and $0.2$, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is:

Three numbers are chosen at random without replacement from $\{1,2,3,\dots ,8\}$. The probability that their minimum is 3, given that their maximum is 6, is:

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is:

If $C$ and $D$ are two events such that $C\subset D$ and $P(D)\ne 0$, then the correct statement among the following is:

Consider $5$ independent Bernoulli'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:

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is:

There are two urns. Urn $A$ has $3$ distinct red balls and urn $B$ has $9$ distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is:

Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\dots ,20\}$.Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$.Statement-2: If the four chosen numbers form an AP, then the set of all possible values of common difference is $\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\}$.

One ticket is selected at random from $50$ tickets numbered $00,01,02,\dots ,49$. Then the probability that the sum of the digits on the selected ticket is $8$, given that the product of these digits is zero, equals:

In a binomial distribution $B(n,p=1/4)$, if the probability of at least one success is greater than or equal to $9/10$, then $n$ is greater than:

A die is thrown. Let $A$ be the event that the number obtained is greater than $3$. Let $B$ be the event that the number obtained is less than $5$. Then $P(A\cup B)$ is:

It is given that the events $A$ and $B$ are such that $P(A)=\frac{1}{4}$, $P(B|A)=\frac{1}{2}$ and $P(A|B)=\frac{2}{3}$. Then $P(B)$ is:

The mean and variance of a random variable having a binomial distribution are 4 and 2 respectively, then $P(X=1)$ is:

Events $A,B,C$ are mutually exclusive events such that $P(A)=\frac{3x+1}{3}$, $P(B)=\frac{1-x}{4}$ and $P(C)=\frac{1-2x}{2}$. The set of possible values of $x$ are in the interval:

Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is:

Let $A$ and $B$ be two events such that $P(\bar{A}\cup \bar{B})=1/6$, $P(A\cap B)=1/4$ and $P(\bar{A})=1/4$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are:

A random variable $X$ has Poisson distribution with mean $2$. Then $P(X>1.5)$ equals:

The number of ways of distributing $8$ identical balls in $3$ distinct boxes so that none of the boxes is empty is