For a loaded die, the probabilities of outcomes are given as under

$$P(1)=P(2)=0.2, P(3)=P(5)=P(6)=0.1 \text { and } P(4)=0.3 \text {. }$$

The die is thrown two times. Let $A$ and $B$ be the events, 'same number each time' and 'a total score is 10 or more', respectively. Determine whether or not $A$ and $B$ are independent.

Refer to question 1 above. If the die were fair, determine whether or not the events $A$ and $B$ are independent.

The probability that atleast one of the two events $A$ and $B$ occurs is 0.6 . If $A$ and $B$ occur simultaneously with probability 0.3 , evaluate $P(\bar{A})+P(\bar{B})$

A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that atleast one of the three marbles drawn be black, if the first marble is red?

Two dice are thrown together and the total score is noted. The events $E$, $F$ and $G$ are 'a total of 4 ', 'a total of 9 or more' and 'a total divisible by 5 ', respectively. Calculate $P(E), P(F)$ and $P(G)$ and decide which pairs of events, if any are independent.

Explain why the experiment of tossing a coin three times is said to have Binomial distribution.

If $A$ and $B$ are two events such that $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}$ and $P(A \cap B)=\frac{1}{4}$, then find

(i) $P(A / B)$.

(ii) $P(B / A)$.

(iii) $P\left(A^{\prime} / B\right)$.

(iv) $P\left(A^{\prime} / B^{\prime}\right)$.

Three events A, B and C have probabilities $\frac{2}{5}, \frac{1}{3}$ and $\frac{1}{2}$, respectively. If, $P(A \cap C)=\frac{1}{5}$ and $P(B \cap C)=\frac{1}{4}$, then find the values of $P(C / B)$ and $P\left(A^{\prime} \cap C^{\prime}\right)$.

Let $E_1$ and $E_2$ be two independent events such that $P\left(E_1\right)=P_1$ and $P\left(E_2\right)=P_2$. Describe in words of the events whose probabilities are

(i) $P_1 P_2$ (ii) $\left(1-P_1\right) P_2$ (iii) $1-\left(1-P_1\right)\left(1-P_2\right)$ (iv) $P_1+P_2-2 P_1 P_2$

A discrete random variable X has the probability distribution as given below

(i) Find the value of $k$.

(ii) Determine the mean of the distribution.

Prove that

(i) $P(A)=P(A \cap B)+P(A \cap \bar{B})$

(ii) $P(A \cup B)=P(A \cap B)+P(A \cap \bar{B})+P(\bar{A} \cap B)$

If $X$ is the number of tails in three tosses of a coin, then determine the standard deviation of $X$.

In a dice game, a player pays a stake of ₹ 1 for each throw of a die. She receives ₹ 5 , if the die shows a 3 , ₹ 2 , if the die shows a 1 or 6 and nothing otherwise, then what is the player's expected profit per throw over a long series of throws?

Three dice are thrown at the same time. Find the probability of getting three two's, if it is known that the sum of the numbers on the dice was six.

Suppose 10000 tickets are sold in a lottery each for ₹ 1 . First prize is of ₹ 3000 and the second prize is of ₹ 2000 . There are three third prizes of ₹ 500 each. If you buy one ticket, then what is your expectation?

A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Bag I contains 3 black and 2 white balls, bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then, another ball is drawn at random. What is the probability of second ball being blue?

Four cards are successively drawn without replacement from a deck of 52 playing cards. What is the probability that all the four cards are king?

If a die is thrown 5 times, then find the probability that an odd number will come up exactly three times.

If ten coins are tossed, then what is the probability of getting atleast 8 heads?

The probability of a man hitting a target is 0.25 . If he shoots 7 times, then what is the probability of his hitting atleast twice?

A lot of 100 watches is known to have 10 defective watches. If 8 watches are selected (one by one with replacement) at random, then what is the probability that there will be atleast one defective watch?

Consider the probability distribution of a random variable $X$.

Calculate

(i) $V\left(\frac{X}{2}\right)$. (ii) Variance of $X$.

The probability distribution of a random variable $X$ is given below

(i) Determine the value of $k$.

(ii) Determine $P(X \leq 2)$ and $P(X>2)$.

(iii) Find $P(X \leq 2)+P(X>2)$.

For the following probability distribution determine standard deviation of the random variable $X$.

A biased die is such that $P(4)=\frac{1}{10}$ and other scores being equally likely. The die is tossed twice. If $X$ is the 'number of fours seen', then find the variance of the random variable $X$.

A die is thrown three times. Let $X$ be the 'number of twos seen', find the expectation of $X$.

Two biased dice are thrown together. For the first die $P(6)=\frac{1}{2}$, the other scores being equally likely while for the second die $P(1)=\frac{2}{5}$ and the other scores are equally likely. Find the probability distribution of 'the number of one's seen'.

Two probability distributions of the discrete random variables $X$ and $Y$ are given below.

Prove that $E\left(Y^2\right)=2 E(X)$.

A factory produces bulbs. The probability that any one bulb is defective is $\frac{1}{50}$ and they are packed in 10 boxes. From a single box, find the probability that

(i) none of the bulbs is defective.

(ii) exactly two bulbs are defective.

(iii) more than 8 bulbs work properly.

Suppose you have two coins which appear identical in your pocket. You know that, one is fair and one is 2 headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?

Suppose that $6 \%$ of the people with blood group 0 are left handed and $10 \%$ of those with other blood groups are left handed, $30 \%$ of the people have blood group 0 . If a left handed person is selected at random, what is the probability that he/she will have blood group 0 ?

If two natural numbers $r$ and $s$ are drawn one at a time, without replacement from the set $S=\{1,2,3, \ldots n\}$, then find $P(r \leq p / s \leq p)$, where $p \in S$.

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.

The random variable $X$ can take only the values $0,1,2$. If

$$P(X=0)=P(X=1)=p \text { and } E\left(X^2\right)=E[X]$$

then find the value of $p$.

Find the variance of the following distribution.

$A$ and $B$ throw a pair of dice alternately. A wins the game, if he gets a total of 6 and $B$ wins, if she gets a total of 7 . If $A$ starts the game, then find the probability of winning the game by $A$ in third throw of the pair of dice.

Two dice are tossed. Find whether the following two events $A$ and $B$ are independent $A=\{(x, y): x+y=11\}$ and $B=\{(x, y): x \neq 5\}$, where $(x, y)$ denotes a typical sample point.

Q. 40 An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with $k$ additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on $k$.

Three bags contain a number of red and white balls as follows Bag I: 3 red balls, Bag II : 2 red balls and 1 white ball and Bag III : 3 white balls. The probability that bag i will be chosen and a ball is selected from it is $\frac{i}{6}$, where $i=1,2,3$. What is the probability that

(i) a red ball will be selected?

(ii) a white ball is selected?

Refer to question 41 above. If a white ball is selected, what is the probability that it came from

(i) Bag II?

(ii) Bag III?

A shopkeeper sells three types of flower seeds $A_1, A_2$ and $A_3$. They are sold as a mixture, where the proportions are $4: 4: 2$, respectively. The germination rates of the three types of seeds are $45 \%, 60 \%$ and $35 \%$. Calculate the probability̌/p>

(i) of a randomly chosen seed to germinate.

(ii) that it will not germinate given that the seed is of type $A_3$.

(iii) that it is of the type $A_2$ given that a randomly chosen seed does not germinate.

A letter is known to have come either from "TATA NAGAR' or from 'CALCUTTA'. On the envelope, just two consecutive letters TA are visible. What is the probability that the letter came from 'TATA NAGAR'?

There are two bags, one of which contains 3 black and 4 white balls while the other contains 4 black and 3 white balls. A die is thrown. If it shows up 1 or 3 , a ball is taken from the Ist bag but it shows up any other number, a ball is chosen from the II bag. Find the probability of choosing a black ball.

There are three urns containing 2 white and 3 black balls, 3 white and 2 black balls and 4 white and 1 black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.

By examining the chest X-ray, the probability that TB is detected when a person is actually suffering is 0.99 . The probability of an healthy person diagnosed to have TB is 0.001 . In a certain city, 1 in 1000 people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?

An item is manufactured by three machines $A, B$ and $C$. Out of the total number of items manufactured during a specified period, $50 \%$ are manufactured on $A, 30 \%$ on $B$ and $20 \%$ on $C$. $2 \%$ of the items produced on $A$ and $2 \%$ of items produced on $B$ are defective and $3 \%$ of these produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine $A$ ?

Let $X$ be a discrete random variable whose probability distribution is defined as follows.

$$P(X=x)= \begin{cases}k(x+1), & \text { for } x=1,2,3,4 \\ 2 k x, & \text { for } x=5,6,7 \\ 0, & \text { otherwise }\end{cases}$$

where, $k$ is a constant. Calculate

(i) the value of $k$.

(ii) $E(X)$.

(iii) standard deviation of $X$.

The probability distribution of a discrete random variable $X$ is given as under

Calculate

(i) the value of $A$, if $E(X)=2.94$.

(ii) variance of $X$.

The probability distribution of a random variable $x$ is given as under

$$P(X=x)=\left\{\begin{array}{l} k x^2, x=1,2,3 \\ 2 k x, x=4,5,6 \\ 0, \text { otherwise } \end{array}\right.$$

where, $k$ is a constant. Calculate

(i) $E(X)$

(ii) $E\left(3 X^2\right)$

(iii) $P(X \geq 4)$

A bag contains $(2 n+1)$ coins. It is known that $n$ of these coins have a head on both sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is $\frac{31}{42}$, then determine the value of $n$.

Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable $X$, where $X$ is the number of aces.

A die is tossed twice. If a 'success' is getting an even number on a toss, then find the variance of the number of successes.

There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let $X$ denotes the sum of the numbers on two cards drawn. Find the mean and variance of $X$.

If $P(A)=\frac{4}{5}$ and $P(A \cap B)=\frac{7}{10}$, then $P(B / A)$ is equal to

If $P(A \cap B)=\frac{7}{10}$ and $P(B)=\frac{17}{20}$, then $P(A / B)$ equals to

If $P(A)=\frac{3}{10}, P(B)=\frac{2}{5}$ and $P(A \cup B)=\frac{3}{5}$, then $P(B / A)+P(A / B)$ equals to

If $P(A)=\frac{2}{5}, P(B)=\frac{3}{10}$ and $P(A \cap B)=\frac{1}{5}$, then $P\left(A^{\prime} / B^{\prime}\right) \cdot P\left(B^{\prime} / A^{\prime}\right)$ is equal to

If $A$ and $B$ are two events such that $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}$ and $P(A / B)=\frac{1}{4}$, then $P\left(A^{\prime} \cap B^{\prime}\right)$ equals to

If $P(A)=0.4, P(B)=0.8$ and $P(B / A)=0.6$, then $P(A \cup B)$ is equal to

If $A$ and $B$ are two events and $A \neq \phi, B \neq \phi$, then

If $A$ and $B$ are events such that $P(A)=0.4, P(B)=0.3$ and $P(A \cup B)=0.5$, then $P\left(B^{\prime} \cap A\right)$ equals to

If $A$ and $B$ are two events such that $P(B)=\frac{3}{5}, P(A / B)=\frac{1}{2}$ and $P(A \cup B)=\frac{4}{5}$, then $P(A)$ equals to

In question 64 (above), $P\left(B / A^{\prime}\right)$ is equal to

If $\quad P(B)=\frac{3}{5}, \quad P(A / B)=\frac{1}{2} \quad$ and $\quad P(A \cup B)=\frac{4}{5}, \quad$ then $P(A \cup B)^{\prime}+P\left(A^{\prime} \cup B\right)$ is equal to

If $P(A)=\frac{7}{13}, P(B)=\frac{9}{13}$ and $P(A \cap B)=\frac{4}{13}$, then $P\left(A^{\prime} / B\right)$ is equal to

If $A$ and $B$ are such events that $P(A)>0$ and $P(B) \neq 1$, then $P\left(A^{\prime} / B^{\prime}\right)$ equals to

If $A$ and $B$ are two independent events with $P(A)=\frac{3}{5}$ and $P(B)=\frac{4}{9}$, then $P\left(A^{\prime} \cap B^{\prime}\right)$ equals to

If two events are independent, then

If $A$ and $B$ be two events such that $P(A)=\frac{3}{8}, P(B)=\frac{5}{8}$ and $P(A \cup B)=\frac{3}{4}$, then $P(A / B) \cdot P\left(A^{\prime} / B\right)$ is equal to

If the events $A$ and $B$ are independent, then $P(A \cap B)$ is equal to

Two events $E$ and $F$ are independent. If $P(E)=0.3$ and $P(E \cup F)=0.5$, then $P(E / F)-P(F / E)$ equals to

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, them the probability of getting exactly one red ball is

Refer to question 74 above. If the probability that exactly two of the three balls were red, then the first ball being red, is

Three persons $A, B$ and $C$, fire at a target in turn, starting with $A$. Their probability of hitting the target are $0.4,0.3$ and 0.2 , respectively. The probability of two hits is

Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has atleast one girl is

If a die is thrown and a card is selected at random from a deck of 52 playing cards, then the probability of getting an even number on the die and a spade card is

A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then probability that both are dead is

If eight coins are tossed together, then the probability of getting exactly 3 heads is

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

Which one is not a requirement of a Binomial distribution?

If two cards are drawn from a well shuffled deck of 52 playing cards with replacement, then the probability that both cards are queens, is

The probability of guessing correctly atleast 8 out of 10 answers on a true false type examination is

If the probability that a person is not a swimmer is 0.3 , then the probability that out of 5 persons 4 are swimmers is

The probability distribution of a discrete random variable $X$ is given below

The value of k is

For the following probability distribution.

E(X) is equal to

For the following probability distribution.

$E\left(X^2\right)$ is equal to

Suppose a random variable $X$ follows the Binomial distribution with parameters $n$ and $p$, where $0

In a college, $30 \%$ students fail in Physics, $25 \%$ fail in Mathematics and $10 \%$ fail in both. One student is chosen at random. The probability that she fails in Physics, if she has failed in Mathematics is

$A$ and $B$ are two students. Their chances of solving 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 a box has 100 pens of which 10 are defective, then what is the probability that out of a sample of 5 pens drawn one by one with replacement atmost one is defective?

If $P(A)>0$ and $P(B)>0$. Then, $A$ and $B$ can be both mutually exclusive and independent.

If $A$ and $B$ are independent events, then $A^{\prime}$ and $B^{\prime}$ are also independent.

If $A$ and $B$ are mutually exclusive events, then they will be independent also.

Two independent events are always mutually exclusive.

If $A$ and $B$ are two independent events, then $P(A$ and $B)=P(A) \cdot P(B)$.

Another name for the mean of a probability distribution is expected value.

If $A$ and $B^{\prime}$ are independent events, then $P\left(A^{\prime} \cup B\right)=1-P(A) P\left(B^{\prime}\right)$.

If A and B are independent, then P (exactly one of $\mathrm{A}, \mathrm{B}$ occurs)

$$=P(A) P\left(B^{\prime}\right)+P(B) P\left(A^{\prime}\right) \text {. }$$

If $A$ and $B$ are two events such that $P(A)>0$ and $P(A)+P(B)>1$, then $P(B / A) \geq 1-\frac{P\left(B^{\prime}\right)}{P(A)}$

If $A, B$ and $C$ are three independent events such that

$$P(A)=P(B)=P(C)=p,$$

then $P$ (atleast two of $A, B$ and $C$ occur) $=3 p^2-2 p^3$.

If $A$ and $B$ are two events such that $P(A / B)=p, P(A)=p, P(B)=\frac{1}{3}$ and $P(A \cup B)=\frac{5}{9}$, then $p$ is equal to ............... .

If $A$ and $B$ are such that $P\left(A^{\prime} \cup B^{\prime}\right)=\frac{2}{3}$ and $P(A \cup B)=\frac{5}{9}$, then $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)$ is equal to ............ .

If $X$ follows Binomial distribution with parameters $n=5, p$ and $P(X=2)=9 P(X=3)$, then $p$ is equal to ............. .

If X be a random variable taking values $x_1, x_2, x_3, \ldots, x_{\mathrm{n}}$ with probabilities $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \ldots, \mathrm{P}_{\mathrm{n}}$, respectively. Then, $\operatorname{Var}(x)$ is equal to ......... .

Let $A$ and $B$ be two events. If $P(A / B)=P(A)$, then $A$ is ........... of $B$.