Random Experiment

Q.

Consider the experiment of throwing a die, if a multiple of 3 comes up throw the die again and if any other number comes, toss a coin. Find the conditional probability fo the event the coin shows a tail, given that atleast one die shows a 3.

Q. The number of $5$ letters palindrome words that can be formed only using vowels is

1. $24$
2. $25$
3. $125$
4. $250$

Q. In a class there are $10$ men and $20$ women. Out of them half of the number of men and half of the number of women have brown eyes. Out of them if a person is chosen at random, the chance that for the person chosen to be a man or brown eyed person is

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{3}{4}$
4. $\frac{1}{4}$

Q.

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is

(A) 0 (B) (C) (D)

Q. A number is chosen at random from the numbers 10 to 99. By seeing the number a man will laugh if product of the digits is 12. If he choose three numbers with replacement, then the probability that he will laugh atleast once is

1. $1-{\left(\frac{42}{43}\right)}^3$
2. $1-{\left(\frac{31}{45}\right)}^3$
3. $1-{\left(\frac{43}{45}\right)}^3$
4. $1-{\left(\frac{41}{45}\right)}^3$

Q.

The probability that a student is not a swimmer is 1/5. The probability that out of five students, four are swimmers is

(a) ${}^5{C}_4{\left(\frac{4}{5}\right)}^4\frac{1}{5}\left(b\right){\left(\frac{4}{5}\right)}^4\frac{1}{5}\left(c\right){}^5{C}_1\frac{1}{5}{\left(\frac{4}{5}\right)}^4$

(d) None of these

Q.

An event has odds in favour 4 : 5, then the probability that event occurs, is

1. 
   

2. 
   

3. 
   

4. 
   

Q.

In an examination, 20 questions of true-false typer are asked. Suppose a students tosses a fair coin to determine his answer to each question. If the coin falls heads, he answer true, if it falls tails, he answer false. Find the probability that he answers atleast 12 questions correctly,

Q. If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the from $7^m+7^n$ is divisible by 5, equals

1. $\frac{1}{4}$
2. $\frac{1}{7}$
3. $\frac{1}{49}$
4. $\frac{1}{8}$

Q. In India, if $60\%$ population likes tea, $10\%$ likes coffee and $5\%$ likes both. A person is selected at random, then the probability that he(she) does not like both is

1. $35\%$
2. $20\%$
3. $25\%$
4. $30\%$

Q.

A black and a red die are rolled.
Find the conditional probability of obtaining the sum 8 given that the red die resulted in a number less than 4.

Q. Drawing a card from a well shuffled ordinary deck of $52$ playing cards: the events ''card drawn is sapde'' and ''card drawn is ace'' are

1. Mutually exclusive
2. Equally likely
3. Forming an exhaustive system
4. None of these

Q. A coin is tossed then a dice is thrown. Let $A$ be the event ''head turns up on the coin and odd number turns up on the dice'' and $B$ be the event ''tail turns up on the coin and an even number turns up on the dice''. Then $A\cap B^{\prime }$ is
(where $H$ represents head and $T$ represents tail)

1. $\{H1,H3,H5\}$
2. $\{T2,T4,T6\}$
3. $\varphi$
4. $\{H1,H2,H3,H4,H5,H6,T1,T3,T5\}$

Q.

Trials of a random experiment are called Bernoulli trials, if they satisfy the condition/s:

1. There should be a finite number of independent trials

2. Each trial has exactly two outcomes: success or failure.

3. The probability of success remains the same in each trial.

4. All of the these

Q. A natural number $X$ is chosen at random from the first $120$ natural numbers and it is observed that it is divisible by $8$, then the probability that it is not divisible by $6$ is:

1. $\frac{1}{3}$
2. $\frac{1}{4}$
3. $\frac{3}{4}$
4. $\frac{2}{3}$

Q. In a bag, there are 6 balls of which 3 are white and 3 are black. 6 balls are drawn successively $\left(i\right)$ without replacement, $\left(ii\right)$ with replacement. What is the chance that the colors are alternate?

1. $\frac{1}{20}$; $\frac{1}{64}$
2. $\frac{1}{10}$; $\frac{1}{64}$
3. $\frac{1}{20}$; $\frac{1}{32}$
4. $\frac{1}{10}$; $\frac{1}{32}$

Q.

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Q.

Three coins are tossed once. Find the probability of getting

(i) 3 heads (ii) 2 heads (iii) at least 2 heads

(iv) at most 2 heads (v) no head (vi) 3 tails

(vii) exactly two tails (viii) no tail (ix) at most two tails.

Q. Let $E$ denote the set of letters of the English alphabet, $V=\{a,e,i,o,u\}$, and $C$ be the complement of $V$ in $E$. Then, the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is

1. $261870$
2. $314160$
3. $425880$
4. $851760$

Q. Let $X$ denotes the sum of the numbers obtained when two fair dice are rolled. Then the standard deviation of $X$ is:

1. $1.414$
2. $2.415$
3. $1.715$
4. $1.2$

Q. The number of odd numbers greater than $8000$ that can be formed using the digits $2,3,4,5$ and $8$, if repetition is not allowed is

1. $60$
2. $65$
3. $24$
4. $12$

Q. In a random experiment with sample space $S,$ two events $A$ and $B$ are said to be equally likely, if

1. $A=B$
2. $n\left(A\right)=n\left(B\right)$
3. $A\cap B=\varphi$
4. $A\cup B=S$

Q.

A showroom has 10 different pairs of expensive shoes. A thief steals 4 shoes randomly from the showroom.
Match the following:
$\begin{array}{cccc}& Column-I& & ColumnII\\ \left(a\right)& Theprobabilitythatthereisno& \left(p\right)& \frac{3}{19\times 17}\\ & matchingpairamongthe4shoesstolenis& & \\ \left(b\right)& Theprobabilitythatthereisatleastone& \left(q\right)& \frac{16}{19\times 17}\\ & matchingpairamongthe4shoesstolenis& & \\ \left(c\right)& Theprobabilitythatthereisexactlyone& \left(r\right)& \frac{{}^{20}{C}_4{-}^{{}^{10}{C}_4{.2}^4}}{{}^{20}{C}_4}\\ & matchingpairamongthe4shoesstolenis& & \\ \left(d\right)& Theprobabilitythattherearetwomatching& \left(s\right)& \frac{14\times 16}{19\times 17}\\ & pairsamongthe4shoesstolenis& & \\ & & \left(t\right)& \frac{98}{19\times 103}\end{array}$

1. (A) $\to$ (q) (B) $\to$ (s) (C) $\to$ (p) (D) $\to$ (r)
2. (A) $\to$ (r) (B) $\to$ (s) (C) $\to$ (p) (D) $\to$ (q)
3. (A) $\to$ (s) (B) $\to$ (r) (C) $\to$ (t) (D) $\to$ (p)
4. (A) $\to$ (s) (B) $\to$ (r) (C) $\to$ (q) (D) $\to$ (p)

Q. A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes. [NCERT EXEMPLAR]

Q. Six boys and six girls sit in a row randomly. The probability that six girls sit together is

Q. In a class of $70$ students there are $40$ boys and $30$ girls. $20$ percent of boys and $30$ percent of girls have grey hair and remaining have black hair. A student is selected at random from the class. Let $A,B$ denote the event that the selected student has grey hair, black hair repectively and $E,F$ denote the event that the selected student is boy, girl repectively, then $P(B\cap F)=$

1. $\frac{8}{17}$
2. $\frac{3}{10}$
3. $\frac{3}{7}$
4. $\frac{7}{10}$

Q. A coin is tossed. If it shows a tail, we draw a ball from a box which contains $2$ red and $3$ black balls. If it shows head, we throw a die. Find the sample space for this experiment.

Q. A coin is tossed three times, consider the following events:
$A:$ no head appears
$B:$ exactly one head appears
$C:$ at least two heads appears
Which of the following is/are true?

1. $A,B$ and $C$ are exhaustive events
2. $A$ and $C$ are mutually exclusive events
3. $B$ and $C$ are mutually exclusive events
4. $A,B$ and $C$ are pairwise mutually exclusive events

Q. A cricket club has $15$ members, of whom only $5$ can bowl. If the names of $15$ members are put into a box and $11$ are drawn at arandom, then the probability of getting an eleven containing at least $3$ bowlers is

1. $\frac{7}{13}$
2. $\frac{6}{13}$
3. $\frac{11}{15}$
4. $\frac{12}{13}$

Q. A coin is tossed and a die is thrown. Let $A$ be the event 'Head turns up on the coin and an odd number turns up on the die' and $B$ be the event 'Tail turns up on the coin and an even number turns up on the die'. Which of the following is/are true:

1. $n(A\cap B)=6$
2. $n(A\cup B)=6$
3. $n(A\cap B)=0$
4. $n(A\cup B)=0$