Probability practice · Mixed difficulty

A number $x$ is chosen at random from the set $\{1,2,3, \ldots, 100\}$. Let $p=$ probability that $x$ is divisible by 19 and $q=$ probability that $x$ is divisible by 31, then $p+q$ is equal to

Three-digit numbers are formed using the digits $0 ,$ $1,2,3,4,5$ without repetition of digits. If a number is chosen at random, then the probability that the digits either increase or decrease, is

The probability that in a year of 22 nd century chosen at random, there will be 53 Sundays, is

If $n\left( A \right) = 8$ and $n\left( {A \cap B} \right) = 2,$ then $n\left[ {{{\left( {A \cap B} \right)}^\prime } \cap A} \right]$ is equal to

Suppose $f\left( x \right) = {x^3} + a{x^2} + bx + c,$ where $a,\,b,\,c$ are chosen respectively by throwing a dice three times. Then, the probability that $f\left( x \right)$ is an increasing function, is

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is

The probability that in the random arrangement of the letters of the word 'UNIVERSITY', the two I's does not come together is

For the 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 events occur simultaneously) $= {p^2},\;$ where $0 < p < \frac{1}{2}$. Then, the probability of at least one of the three events $A$, $B$ and $C$ occuring, is

The probability that the same number appear on throwing three dice simultaneously, is

Two dice are thrown. If it is known that the sum of the numbers on the dice is less than 6 , the probability of getting a sum 3 is

$A$ and $B$ are two students. Their chances of sloving a problem correctly are $\frac{1}{3}$ and $\frac{1}{4}$, respectively. If the probability of their making a common error is, $\frac{1}{20}$ and they obtain the same answer, then the probability of their answer to be correct is

If three six-faced fair dice are thrown together, the probability that the sum of the numbers appearing on the dice is 16 is

$A$ and $B$ are two independent events such that $P\left( A \right)\frac{1}{2}$ and $P\left( B \right) = \frac{1}{3},\;$ then $P$ (neither $A$ nor $B$) is equal to

5 persons $A, B, C, D$ and $E$ are in a queue of a shop. The probability that $A$ and $E$ always occur together, is

Consider the system of equations $\mathrm{px}+\mathrm{qy}=0, \mathrm{rx}+\mathrm{sy}=0$ where $p, q, r, s \in\{-1,1\}$
Statement-1: The probability that the system of equation has a unique solution is $\frac{1}{2}$.
Statement-2: The probability that the system of equations no solution is zero.

A coin is tossed 10 times. The probability of getting exactly six heads is

Two dice are tossed once. The probability of getting even number at the first die or a total of 8 is

Suppose that the probability that an item produced by a particular machine is defective equals 0.2. If 10 items produced from this machine are selected at random, the probability that not more than one defective is found is

A bag contains 6 red and 3 white balls. Four balls are drawn one by one and not replaced. The probability that they are alternatively of different colours, is

A number $n$ is randomly selected from the set $\{1,2,3, \ldots ., 1000\}$. The probability that $\frac{\sum_{i=1}^{n} i^{2}}{\sum_{i=1}^{n} i}$ is an integer is

An investment consultant predicts that the odds ageinst the price of a certain stock will go up during the next week are $2: 1$ and the odds in favour of the price remaining the same are $1: 3$. The probability that the price of the stock will go down during the next week, is

In a class of 125 students 70 passed in Mathematics, 55 in Statistic and 30 in both. The probability that a student selected at random from the class, has passed in only one subject is

There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i, i=1,2,3,4,5$.But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and none of the remaining students gets the seat previously allotted to him/her, is

The letters of the word 'longest' are arranged at random. The probability that the vowels may occupy only odd positions, is

It has been found that if $A$ and $B$ play a game 12 times, $A$ wins 6 times, $B$ wins 4 times and they draw twice. $A$ and $B$ take part in a series of 3 games. The probability that they will win alternately is

Two cards are drawn successively with replacement from a well shuffled deck of 52 cards, then the mean of the number of aces is

Three rifle-men take one shot each at the same target. The probability of the first rifle-man hitting the target is 0.4 , the probability of the second rifle-man hitting the target is 0.5 and the probability of the third rifle-man hitting the target is 0.8 . The Probability that exactly two of them hit the target, is

The odds against a certain event are 5 :2 and the odds in favour of another independent event are 6 : 5. The probability that at least one of the events will happen, is

A man takes a step forward with probability $0.4$ and backwards with probability 0.6. The probability that at the end of eleven steps he is just one step away from the starting point, is

A bag contains 10 mangoes out of which 4 are rotten, two mangoes are taken out together. If one of them is found to be good, the probability that other is also good is