Probability

Q104 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………….. FillBlank Q105 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………. FillBlank Q106 If X follows Binomial distribution with parameters ${\rm{n}} = 5,$ ${\rm{p}}$ and $P(X = 2) = 9P(X = 3)$, then $p$ is equal to…………… FillBlank Q107 If X be a random variable taking values ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ with probabilities ${{\rm{P}}_1},{{\rm{P}}_2},{{\rm{P}}_3}, \ldots ,{{\rm{P}}_{\rm{n}}}$, respectively. Then, Var$(x)$ is equal to……………. FillBlank Q108 Let $A$ and $B$ be two events. If $P(A/B) = P(A)$, then A is ………… of B. FillBlank Q41 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? LA Q42 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? LA Q43 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 probability

(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. LA Q44 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'? LA Q45 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. LA Q46 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. LA Q47 By examining the chest X-ray, the probability that ${\rm{TB}}$ 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? LA Q48 An item is manufactured by three machines $A,{\rm{ }}B and C$. 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 $A$ ? LA Q49 Let $X$ be a discrete random variable whose probability distribution is defined as follows.

$P(X = x) = \left\{ {\begin{array}{llllllllllllllllllll}{k(x + 1),}&{{\rm{ for }}x = 1,2,3,4}\\{2kx,}&{{\rm{ for }}x = 5,6,7}\\{0,}&{{\rm{ otherwise }}}\end{array}} \right.$

where, $k$ is a constant. Calculate

(i) the value of $k$.

(ii) $E(X)$.

(iii) standard deviation of $X$. LA Q50 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 $X$. LA Q51 The probability distribution of a random variable $x$ is given as under

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

where, $k$ is a constant. Calculate

(i) $E(X)$

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

(iii) $P(X \ge 4)$ LA Q52 A bag contains $(2n + 1)$ 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 $n$. LA Q53 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. LA Q54 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. LA Q55 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 $X$. LA Q56 If $P(A) = \frac{4}{5}$ and $P(A \cap B) = \frac{7}{{10}}$, then $P(B/A)$ is equal to MCQ Q57 If $P(A \cap B) = \frac{7}{{10}}$ and $P(B) = \frac{{17}}{{20}}$, then $P(A/B)$ equals to MCQ Q58 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 MCQ Q59 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 MCQ Q60 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 MCQ Q61 If $P(A) = 0.4,P(B) = 0.8$ and $P(B/A) = 0.6$, then $P(A \cup B)$ is equal to MCQ Q62 If A and B are two events and $A \ne \phi ,B \ne \phi$, then MCQ Q63 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 MCQ Q64 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 MCQ Q65 In question 64 (above), $P\left( {B/{A^\prime }} \right)$ is equal to MCQ Q66 If $P(B) = \frac{3}{5},P(A/B) = \frac{1}{2}$ and $P(A \cup B) = \frac{4}{5}$, then
$P{(A \cup B)^\prime } + P\left( {{A^\prime } \cup B} \right)$ is equal to MCQ Q67 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 MCQ Q68 If A and B are such events that $P(A) > 0$ and $P(B) \ne 1$, then $P\left( {{A^\prime }/{B^\prime }} \right)$ equals to MCQ Q69 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 MCQ Q70 If two events are independent, then MCQ Q71 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 MCQ Q72 If the events A and B are independent, then $P(A \cap B)$ is equal to MCQ Q73 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 MCQ Q74 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 MCQ Q75 Refer to question 74 above. If the probability that exactly two of the three balls were red, then the first ball being red, is MCQ Q76 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 MCQ Q77 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 MCQ Q78 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 MCQ Q79 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 MCQ Q80 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 MCQ Q81 If eight coins are tossed together, then the probability of getting exactly 3 heads is MCQ Q82 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 MCQ Q83 Which one is not a requirement of a Binomial distribution? MCQ Q84 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 MCQ Q85 The probability of guessing correctly atleast 8 out of 10 answers on a true false type examination is MCQ Q86 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 MCQ Q87 The probability distribution of a discrete random variable X is given below

X 2 3 4 5
P(X) $\frac{5}{k}$ $\frac{7}{k}$ $\frac{9}{k}$ $\frac{{11}}{k}$

The value of $k$ is MCQ Q88 For the following probability distribution.
X -4 -3 -2 -1 0
P(X) 0.1 0.2 0.3 0.2 0.2
$E(X)$ is equal to MCQ Q89 For the following probability distribution.
X 1 2 3 4
P(X) $\frac{1}{{10}}$ $\frac{1}{5}$ $\frac{3}{{10}}$ $\frac{2}{5}$
$E\left( {{X^2}} \right)$ is equal to MCQ Q90 Suppose a random variable X follows the Binomial distribution with parameters $n$ and $p$, where $0 < p < 1$. If $P(x = r)/P(x = n - r)$ is independent of $n$ and $r$, then $p$ equals to MCQ Q91 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 MCQ Q92 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 MCQ Q93 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? MCQ Q1 For a loaded die, the probabilities of outcomes are given as under
$P(1) = P(2) = 0.2,$ $P(3) = P(5) = P(6) = 0.1$ and $P(4) = 0.3$.
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. SA Q2 Refer to question 1 above. If the die were fair, determine whether or not the events A and B are independent. SA Q3 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 $P(\bar A) + P(\bar B)$. SA Q4 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? SA Q5 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. SA Q6 Explain why the experiment of tossing a coin three times is said to have Binomial distribution. SA Q7 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) $P\left( {{A^\prime }/{B^\prime }} \right)$ SA Q8 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 $P\left( {{A^\prime } \cap {C^\prime }} \right)$. SA Q9 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) ${P_1} + {P_2} - 2{P_1}{P_2}$ SA Q10 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. SA Q11 Prove that

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

(ii) $P(A \cup B) = P(A \cap B) + P(A \cap \bar B) + P(\bar A \cap B)$ SA Q12 If X is the number of tails in three tosses of a coin, then determine the standard deviation of X. SA Q13 In a dice game, a player pays a stake of Rs.1 for each throw of a die. She receives Rs.5, if the die shows a 3, Rs.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? SA Q14 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. SA Q15 Suppose 10000 tickets are sold in a lottery each for Rs.1. First prize is of Rs.3000 and the second prize is of Rs.2000. There are three third prizes of Rs.500 each. If you buy one ticket, then what is your expectation? SA Q16 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. SA Q17 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. SA Q18 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? SA Q19 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? SA Q20 If a die is thrown 5 times, then find the probability that an odd number will come up exactly three times. SA Q21 If ten coins are tossed, then what is the probability of getting atleast 8 heads? SA Q22 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? SA Q23 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? SA Q24 Consider the probability distribution of a random variable X.

Calculate
(i) $V\left( {\frac{X}{2}} \right)$.
(ii) Variance of X. SA Q25 The probability distribution of a random variable X is given below

(i) Determine the value of $k$.
(ii) Determine $P(X \le 2)$ and $P(X > 2)$.
(iii) Find $P(X \le 2) + P(X > 2)$. SA Q26 For the following probability distribution determine standard deviation of the random variable X.

SA Q27 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. SA Q28 A die is thrown three times. Let X be the 'number of twos seen', find the expectation of X. SA Q29 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' SA Q30 Two probability distributions of the discrete random variables X and Y are given below.

Prove that $E\left( {{Y^2}} \right) = 2E(X)$. SA Q31 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. SA Q32 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? SA Q33 Sup pose that 6\% of the people with blood group O 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 O? SA Q34 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 $P(r \le p/s \le p)$, where $p \in S$. SA Q35 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. SA Q36 The random variable X can take only the values 0,1,2. If
$P(X = 0) = P(X = 1) = p$ and $E\left( {{X^2}} \right) = E[X]$
then find the value of $p$. SA Q37 Find the variance of the following distribution.

SA Q38 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. SA Q39 Two dice are tossed. Find whether the following two events A and B are independent $A = \{ (x,y):x + y = 11\}$ and $B = \{ (x,y):x \ne 5\}$,

where $(x,y)$ denotes a typical sample point. SA Q40 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 $k$. SA

Q1 Given that $E$ and F are events such that $P(E) = 0.6,P(F) = 0.3$ and $P(E \cap P) = 0.2$ , find $P(E|F)$ and $P(F|E)$ . SA Q2 Compute $P(A|B)$ , if $P(B) = 0.5$ and $P(A \cap B) = 0.32$ SA Q3 If $P(A) = 0.8,P(B) = 0.5$ and $P(B|A) = 0.4$ , find

(i) $P(A \cap B)$

(ii) $P(A|B)$

(iii) $P(A \cup B)$ SA Q4 Evaluate $(A \cup B)$ , if $2P(A) = P(B) = \cfrac{5}{{13}}$ and $P(A|B) = \cfrac{2}{5}$ SA Q5 If $P(A) = \cfrac{6}{{11}},\;P(B) = \cfrac{5}{{11}}\;{\rm{and}}\;P(A \cup B) = \cfrac{7}{{11}}$ , find

(i) $P(A \cap B)$

(ii) $P(A|B)$

(iii) $P(B|A)$ SA Q6 A coin is tossed three times, where

(i) $E$ : head on third toss, $F$ : heads on first two tosses

(ii) $E$ : at least two heads, $F$ : at most two heads

(iii) $E$ : at most two tails, $F$ : at least one tail SA Q7 Two coins are tossed once, where

(i) $E$ : tail appears on one coin, $F$ : one coin shows head

(ii) $E$ : no tail appears, $F$ : no head appears SA Q8 A dice is thrown three times,

$E$ : 4 appears on the third toss,

$F$ : 6 and $5$ appears respectively on first two tosses SA Q9 Mother, father and son line up at random for a family picture
$E:$ son on one end, $F:$ father in middle SA Q10 A black and a red dice are rolled.

(a) Find the conditional probability of obtaining a sum greater than 9, given that the black dice resulted in a 5.

(b) Find the conditional probability of obtaining the sum 8, given that the red dice resulted in a number less than 4. SA Q11 A fair dice is rolled. Consider events $E = \{ 1,3,5\} ,F = \{ 2,3\}$ and $G = \{ 2,3,4,5\} .$ Find

(i) $P(E/F)$ and $P(F/E)$

(ii) $P(E/G)$ and $P(G/E)$

(iii) $P((E \cup F)/G)$ and $P((E \cap F)/G)$ SA Q12 Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the condition probability that both are girls given that (i) the youngest is a girl, (ii) at least one is the girl? SA Q13 An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy SA Q14 Given that the two numbers appeaing on throwing two dices are different. Find the probability o the event ‘the sum of numbers on the dice is 4’. SA Q15 Consider the experiment of throwing a dice, if a multiple of 3 comes up, throw the dice again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one dice shows a 3’. SA Q16 If $P(A) = \cfrac{1}{2},P(B) = 0,$ then $P(A|B)$ is

(A) 0

(B) $\cfrac{1}{2}$

(C) not defined

(D) 1 SA Q17 If $A$ and $B$ are events such that $P(A/B) = P(B/A),$ then

(A) $A \subset B$ but $A \ne B$

(B) $A = B$

(C) $A \cap B = \phi$

(D) $P(A) = P(B)$ SA

Q1 If $P(A) = \cfrac{3}{5}$ and $P(B) = \cfrac{1}{5},$ find $P(A \cap B)$ if $A$ and $B$ are independent events. SA Q2 Two cards are drawn at random and without replacement from a puck of 52 playing cards. Find the probability that both the cards are black. SA Q3 A box of oranges is inspected by examining three randomly selected oranges drawn without replacement Ifan 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 be approved for sale. SA Q4 A fair coin and an unbiased dice are tused. Let $A$ be the event head appears on the coin' and $B$ be the event 3 on the dice'. Check whether $A$ and $B$ are independent events or not ? SA Q5 A dice marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let $A$ be the event, the number is even and $B$ be the event, 'the number is red'. Are $A$ and $B$ independent? SA Q6 Let $E$ and $F$ be events with $P(E) = \cfrac{3}{5},P(F) = \cfrac{3}{{10}}$ and $P(E \cap F) = \cfrac{1}{5}.$ Are $E$ and $F$ independent? SA Q7 Given that the events $A$ and $B$ are such that $P(A) = \cfrac{1}{2},P(A \cup B) = \cfrac{3}{5}$ and $P(B) = p.$ Find $p$ if they are (i) mutually exclusive (ii) independent. SA Q8 Let $A$ and $B$ are independent events with $P(A) = 0.3$ and $P(B) = 0.4$ . Find

(i) $P(A \cap B)$

(ii) $P(A \cup B)$

(iii) $P(A|B)$

(iv) $P(B|A)$ SA Q9 If $A$ and $B$ are two events such that $P(A) = \cfrac{1}{4},P(B) = \cfrac{1}{2}$ and $P(A \cap B) = \cfrac{1}{8},$ find $P$ (not $A$ and not $B$ ). SA Q10 Events $A$ and $B$ are such that $P(A) = \cfrac{1}{2},P(B) = \cfrac{7}{{12}}$ and $P({\rm{ not }}A{\rm{ or not }}B) = \cfrac{1}{4}$ State whether $A$ and $B$ are independent? SA Q11 Given two independent events $A$ and $B$ such that $P(A) = 0.3,P(B) = 0.6$ . Find

(i) $P(A{\rm{ and }}B)$

(ii) $P(A{\rm{ and not }}B)$

(iii) $P(A{\rm{ or }}B)$

(iv) $P$ (neither $A$ nor $B$ ) SA Q12 A dice is tossed thrice. Find the probability ofgetting an odd number at least once. SA Q13 Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that

(i) both balls are red.

(ii) first ball is black and second is red.

(iii) one of them is black and other is red

. SA Q14 Probability of solving specific problem independently by $A$ and $B$ are $\cfrac{1}{2}$ and $\cfrac{1}{3}$ respectively. If both try to solve the problem independently, then find the probability that
the problem is solved

(ii) exactly one of them solves the problem. SA Q15 One card is drawn at random from a well shumed dech of 52 cards. In which of the following cases are the events $E$ and $F$ independent?

(i) $E$ : `the car4 Brawn is a spade'
$F$ : `the card drawn is an ace'

(ii) $E$ : `the card drawn is black'
$F$ : `the card drawn is a king'

(iii) $E$ : `the card drawn is a king or queen'

$F$ : `the card drawn is a queen or jack'. SA Q16 In a hostel, 60\% of the students read Hindi newspaper, 40\% read English newspaper and 20\% read both Hindi and English newspapers. A student is selected at random.

(a) Find the probability that she reads neither Hindi nor English newspapers.

(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.

(c) If she reads English newspaper, find the probability that she reads Hindi newspaper. SA Q17 The probability of obtaining an even prime number on each dice when a pair of dice is rolled is

(i) 0

(ii) $\cfrac{1}{3}$

(iii) $\cfrac{1}{{12}}$

(iv) $\cfrac{1}{{36}}$ SA Q18 Two events $A$ and $B$ will be independent, if

(A) $A$ and $B$ are mutually exclusive

(B) $P\left( {{A^\prime } \cap {B^\prime }} \right) = [1 - P(A)][1 - P(B)]$

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

(D) $P(A) + P(B) = 1$ SA

Q1 An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned lo the urn. Moreover, 2 additional balls of the colour drawn are put in lhe urn and then a ball is drawn at random. What is the probability that the second ball is red? SA Q2 A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag. SA Q3 Of the students in a college, it is known that 60\% reside in hostel and 40\% are day scholars (not residing in hostel). Previous year results results report that 30\% of all students who reside in hostel attain A grade and 20 of day scholars attain $A$ grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier? SA Q4 In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\cfrac{3}{4}$ be the probability that he knows the answer and $\cfrac{1}{4}$ the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\cfrac{1}{4}$ . What is the probability that the student knows the answer given that he answered it correctly? SA Q5 A laboratory blood test is 99\% effective in detecting a certain disease when it is in fact, present However, the test also yields a false positive result for 0.5\% of the healthy person tested i.e., if a healthy person is tested, then, with probability 0.005, the test will imply he has the diseases). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive? SA Q6 There are three coins. One is a bvo headed coin (having head on both , another is a biased coin that comes up with heads 75\% of the time and third is an unbiased coin. One ofthe three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin? SA Q7 An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the lnsured persons meets with an accident. What is the probability that he is a scooter driver? SA Q8 A factory has two machines A and B. Past record shows that machine A produced 60\% of the items of output and machine B produced 40\% of the items. Further, 2\% of the items produced by machine A and 1\% produced by machine R were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B? SA Q9 Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, ifthe first group wins, the probability ofintroducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group. SA Q10 Suppose a girl throws a dice. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the dice? SA Q11 A manufacturer has three machine operators A, B and C. The first operator A produces 1\% defective items, where as the other two operators B and C produce 5\% and 7\% defective items respectively. A is on the job for 50\% of the time, B is on the job for 30\% of the time and C is on the job for 20\% of the time. A defective item is produced, what is the probability that it was produced by A? SA Q12 Acard from a pack of 52 card is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond. SA Q13 Probability that $A$ speaks truth is $\cfrac{4}{5}$ . A coin is tossed. $A$ reports that a head appears. The probability that actually there was head is

(A) $\cfrac{4}{5}$

(B) $\cfrac{1}{2}$

(C) $\cfrac{1}{5}$

(D) $\cfrac{2}{5}$ SA Q14 If $A$ and $B$ are two events such that $A \subset B$ and $P(B) \ne 0$ , then which of the following is correct?

(A) $P(A|B) = \cfrac{{P(B)}}{{P(A)}}$

(B) $P(A|B) < P(A)$

(C) $P(A|B) \ge P(A)$

(D) None of these SA

Q1 State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

(i) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$

(ii) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,3\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\, - 0.1\end{array}$

(iii) $\begin{array}{l}Y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.6\,\,\,\,\,\,\,\,\,\,\,\,\,0.1\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$

(iv) $\begin{array}{l}Z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,0\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.3\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\, - 0.1\end{array}$ SA Q2 An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let $X$ represent the number of black balls.
What are the possible values of $X$ ? Is $X$ a random variable? SA Q3 Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of $X$ ? SA Q4 Find the probability distribution of
(i) number of heads in two tosses of a coin.
(ii) number oftails in the simultaneous tosses of three coins.
(iii) number of heads in four tosses of a coin
. SA Q5 Find the probabilty distribution of the number of successes in two tosses of a dice, where a success is defined as
number greater than 4
six appears on at least one dice SA Q6 From alot of 30 bulbs which include 6 defective, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs. SA Q7 A coin is baised so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails. SA Q8 A random variable $X$ has the following probability distribution:

Determine
(i) $k$

(ii) $P(X < 3)$

(iii) $P(X > 6)$

(iv) $P(0 < X < 3)$ SA Q9 The random variable $X$ has a probability distribution $P(X)$ of the following form, where $k$ is some number:

$P(X) = \left\{ {\begin{array}{llllllllllllllllllll}{k,{\rm{ if }}X = 0}\\{2k,{\rm{ if }}X = 1}\\{3k,{\rm{ if }}X = 2}\\{0,{\rm{ other wise }}}\end{array}} \right.$

(a) Determine the value of $k$ .

(b) Find $P(X < 2),P(X \le 2),P(X \ge 2)$ SA Q10 Find the mean number of heads in three tosses of a fair coin. SA Q11 Two dice are thrown simultaneously. If $X$ denotes the number of sixes, find the expectation of $X$ SA Q12 Two numbers are selected at random (without replacement) from the first six positive integers. Let $X$ denote the larger of the two numbers obtained. Find $E(X)$ . SA Q13 Let $X$ denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of $X$ . SA Q14 A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 years. One student is selected in such a manner that each has the same chance of 5 being chosen and the age $X$ of the selected student is recorded. What is the probability distribution of the random variable $X$ ? Find mean, variance and standard deviation of $X$ . SA Q15 In a meeting, $70\%$ of the members favour and $30\%$ oppose a certain proposal. A member is selected at random and we take $X = 0$ if he opposed and $X = 1$ if he is in favour. Find $E(X)$ and ${\mathop{\rm Var}\nolimits} (X)$ . SA Q16 The mean of the numbers obtained on throwing a dice having written 1 on three faces, 2 on two faces and 5 on one face

(A) 1

(B) 2

(C) 5

(D) $\cfrac{8}{3}$ SA Q17 Suppose that two cards are drawn at random from a deck of cards. Let $X$ be the number of aces obtained. Then the value of $E(X)$ is

(A) $\cfrac{{37}}{{221}}$

(B) $\cfrac{5}{{13}}$

(C) $\cfrac{1}{{13}}$

(D) $\cfrac{2}{{13}}$ SA

Q1 A dice is thrown 6 times. If getting an odd number is a success, what is the probability of
(i) 5 successes ?

(ii) at least 5 successes ?

(iii) at most 5 successes ? SA Q2 A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes. SA Q3 There are 5\% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item? SA Q4 Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that

(i) all the five cards are spades?
only 3 cards are spades?

(iii) none is a spade? SA Q6 The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs

(i) none
not more than one

(iii) more than one

(iv) at least one will fuse after 150 days of use. SA Q7 A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0? SA Q8 In an examination, 20 questionl of true-false type are asked. Suppose a student tosses a fair coin lo determine his answer to each question. If the coin falls heads, he answers ‘true’, ifit falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly. SA Q9 Suppose $X$ has a binomial distribution
$B\left( {6,\cfrac{1}{2}} \right)$ . Show that $X = 3$ is the most likely outcome. Hint: $P(X = 3)$ is the maximum among all $\left. {P\left( {{x_i}} \right),{x_i} = 0,1,2,3,4,5,6} \right)$ SA Q10 In a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing? SA Q11 A person buys a lottery tickets in 50 lotteries, in each of which his chance of winning a prize is $\cfrac{1}{{100}}$ . What is the probability that he will win a prize

(a) at least once

(b) exactly once

(c) at least twice? SA Q12 Find the probability of getting 5 exactly twice in 7 throws of a dice. SA Q13 Find the probability of throwing at most 2 sixes in 6 throws of a single dice. SA Q14 It is known that 10\% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective? In each of the following, choose the correct answer: SA Q15 In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

(A) ${10^{ - 1}}$

(B) ${\left( {\cfrac{1}{2}} \right)^5}$

(C) ${\left( {\cfrac{9}{{10}}} \right)^5}$

(D) $\cfrac{9}{{10}}$ SA Q15 The probability that a student is not a swimmer is $\cfrac{1}{5}$ . Then the probability that out of five students, four are swimmer is

(A) $^5{C_4}{\left( {\cfrac{4}{5}} \right)^4}\cfrac{1}{5}$

(B) ${\left( {\cfrac{4}{5}} \right)^4}\cfrac{1}{5}$

(C) $^5{C_1}\cfrac{1}{5}{\left( {\cfrac{4}{5}} \right)^4}$

(D) None of these SA

Q1 $A$ and $B$ are two events such that $P(A) \ne 0$ . Find $P(B/A)$ , if

(i) $A$ is a subset of $B$

(ii) $A \cap B = \phi$ SA Q2 A couple has two children,

(i) Find the probability that both children are males, if it is known that at least one of the children is male.

(ii) Find the probability that both children are females, if it is known that the elder child is a female. SA Q3 Suppose that 5\% of men and 0.25\% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there aare equal number of males and females. SA Q4 Suppose that 90\% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed? SA Q5 An urn contains 25 balls of which 10 balls bear a mark $X$ and the remaining 15 bear a mark
$Y$ . A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that

(i) all will bear $X$ mark.

(ii) not more than 2 will bear $Y$ mark.

(iii) at least one ball will bear $Y$ mark.

(iv) the number of balls with $X$ and $Y$ mark will be equal. SA Q6 In a hurdle race, a player has to cross 10 hurdles. The probability that he clear each hurdle is $$\cfrac{5}{6}$$ . What is the probability that he will knock down fewer than 2 hurdles? SA Q7 A dice is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the dice. SA Q8 If a leap year is selected at random, what is the chance that it will contain 53 tuesdays? SA Q9 An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes. SA Q10 . How many times must a man toss a fair coin so that the probability of having at least one head is more than 90\%? SA Q11 In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair dice is thrown. The man decided to throw a dice thrice but he quits as and when he gets a six. Find the expected value of the amount he wind loses. SA Q12 Suppose we have four boxes $A,B,C$ and $D$ containlng oloured marbles as given below:
Box Marble colour
Red White Black
$A$ 1 6 3
$B$ 6 2 2
$C$ 8 1 1
$D$ 0 6 4
One of the boxes has been selected at random and a single marble is drawn from it If the marble is red, what is the probability that it was drawn from box $A$ ?, box $B$ ?, box $C$ ? SA Q13 Assume that the chances of a patient having a heart attack is 40\%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30\% and perscription of certain drug reduces its chances by 25\%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probabihty that the pabent followed a course of meditation and yoga? SA Q14 If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\cfrac{1}{2}$ ). SA Q15 An electronic assembly consists of two subsystems, say, $A$ and $B$ . From previous testing procedures, the following probabilities are assumed to be known :
$P(A{\rm{ fails }}) = 0.2$
$P(B{\rm{ fails alone }}) = 0.15$
$P(A{\rm{ and }}B{\rm{ fail }}) = 0.15$

Evaluate the following probabilities
(i) $P(A{\rm{ fails }}|B$ has failed)
(ii) $P(A{\rm{ fails a lone }})$ SA Q16 Bag $I$ contain 3 red and 4 black balls and Bag $II$ contains 4 red and 5 black balls. One ball is transferred from Bag $I$ to Bag $II$ and then a ball is drawn from Bag $II$ . The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.Choose the correct answer in each of the following SA Q17 If $A$ and $B$ are two events such that $P(A) \ne 0$ and $P(B|A) = 1,$ then

(A) $A \subset B$

(B ) $B \subset A$

(C) $B = \phi$

(D) $A = \phi$ SA Q18 If $P(A|B) > P(A)$ , then which of the following is correct:

(A) $P(B|A) < P(B)$

(B) $P(A \cap B) < P(A) \cdot P(B)$

(C) $P(B|A) > P(B)$

(D) $P(B|A) = P(B)$ SA Q19 If $A$ and $B$ are any two events such that
$P(A) + P(B) - P(A{\rm{ and }}B) = P(A),$ then

(A) $P(B|A) = 1$

(B) $P(A|B) = 1$

(C) $P(B|A) = 0$

(D) $P(A|B) = 0$ SA