Chapter 24

Three numbers are chosen at random without replacement from \(\\{1,2,3, \ldots, 8]\). The probability that their minimum is 3 , given that their maximum is 6 , is: (A) \(3 / 8\) (B) \(1 / 5\) (C) \(1 / 4\) (D) \(2 / 5\)

A class consists of 80 students, 25 of them are girls. If 10 of the students are rich and 20 of the students are fair complexioned, then the probability of selecting a fair complexioned rich girl from the class (assuming three traits as independent) is (A) \(1 / 10\) (B) \(1 / 32\) (C) \(5 / 512\) (D) \(7 / 512\)

\(A\) 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 (A) \(3 / 5\) (B) 0 (C) 1 (D) \(2 / 5\)

A bag contains \(n+1\) coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is \(\frac{7}{12}\), then the value of \(n\) is(A) 5 (B) 4 (C) 3 (D) 2

The probabilites of three events \(A, B\) and \(C\) are \(P(A)=0.6, P(B)=0.4\) and \(P(C)=0.5 .\) If \(P(A \cup B)\) \(=0.8, P(A \cap C)=0.3, P(A \cap B \cap C)=0.2\) and \(P(A \cup\) \(B \cup C) \geq 0.85\), then (A) \(0.2 \leq P(B \cap C) \leq 0.35\) (B) \(0.5 \leq P(B \cap C) \leq 0.85\) (C) \(0.1 \leq P(B \cap C) \leq 0.35\) (D) none of these

If \(\bar{E}\) and \(\bar{F}\) are the complementary events of events \(E\) and \(F\) respectively and if \(0

The probability that certain electronic component fails when first used is \(0.10 .\) If it does not fail immediately, the probability that it lasts for one year is \(0.99 .\) The probability that a new component will last for one year is (A) \(0.891\) (B) \(0.692\) (C) \(0.92\) (D) none of these

The probability that certain electronic component fails when first used is \(0.10 .\) If it does not fail immediately, the probability that it lasts for one year is \(0.99 .\) The probability that a new component will last for one year is (A) \(0.891\) (B) \(0.692\) (C) \(0.92\) (D) none of these

If \(X\) and \(Y\) are the independent random variables for \(B\left(5, \frac{1}{2}\right)\) and \(B\left(7, \frac{1}{2}\right)\), then \(P(X+Y \geq 1)=\) (A) \(\frac{4095}{4096}\) (B) \(\frac{309}{4096}\) (C) \(\frac{4032}{4096}\) (D) none of these

The probability that a man aged \(x\) years will die in a year is \(p\). The probability that out of \(n\) men \(A_{1}, A_{2}, A_{3}, \ldots\) \(A_{n}\), each aged \(x, A_{1}\) will die and be first to die is (A) \(\frac{1}{n^{2}}\) (B) \(1-(1-p)^{\mathrm{n}}\) (C) \(\frac{1}{n^{2}}\left(1-(1-p)^{n}\right)\) (D) \(\frac{1}{n}\left(1-(1-p)^{n}\right)\).

If two events \(A\) and \(B\) are such that \(P\left(A^{\prime}\right)=0.3, P(B)=\) \(0.4\) and \(P\left(A \cap B^{\prime}\right)=0.5\), then \(P\left(B / A \cup B^{\prime}\right)\) equals (A) \(\frac{3}{4}\) (B) \(\frac{5}{6}\) (C) \(\frac{1}{4}\) (D) \(\frac{3}{7}\)

An elevator starts with \(m\) passengers and stops at \(n\) floors \((m \leq n)\). The probability that no two passengers alight at the same floor is (A) \(\frac{n_{P_{m}}}{m^{n}}\) (B) \(\frac{{ }^{n_{P}} P_{\mathrm{m}}}{n^{\mathrm{m}}}\) (C) \(\frac{n_{m_{m}}}{m^{\text {n }}}\) (D) \(\frac{n_{C_{m}}}{n^{\text {mu }}}\)

If the integers \(m\) and \(n\) are chosen at random between 1 and 100 then the probability that a number of the form \(7^{\mathrm{m}}+7^{\mathrm{a}}\) is divisible by 5 is (A) \(\frac{1}{5}\) (B) \(\frac{1}{7}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{49}\)

Three numbers are chosen at random without reploement from \(\\{1,2, \ldots, 10\\}\). The probability that the minimum of the chosen numbers is 3 , or their maximum is 7 is (A) \(\frac{7}{40}\) (B) \(\frac{5}{40}\) (C) \(\frac{11}{40}\) (D) none of these

Fifteen coupons are numbered \(1,2,3, \ldots 15\). Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on the selected coupon is 9 is (A) \(\left(\frac{9}{16}\right)^{6}\) (B) \(\left(\frac{8}{15}\right)^{7}\) (C) \(\left(\frac{3}{5}\right)^{7}\) (D) none of these

Consider a set ' \(P\) ' containing \(n\) elements. A subset ' \(A\) ' of ' \(P\) ' is drawn and there after set ' \(P^{\prime}\) is reconstructed. Now, one more subset ' \(B\) ' of ' \(P\) ' is drawn. Probabilityof drawing sets \(A\) and \(B\) so that \(A \cap B\) has exactly one element is (A) \((3 / 4)^{\mathrm{n}} \cdot n\) (B) \(n \cdot(3 / 4)^{n-1}\) (C) \(n \cdot(3 / 4)^{n}\) (D) none of these

At a railway station a passenger leaves his luggage in a locker which is opened by dialling a three-digit code (say, \(253,009,325 \ldots\) ). The passenger chooses the code, closes the locker and leaves for the town. A strange man, who does not know the code, tries to open the locker by dialling three digits at random. The probability that the locker opens after \(k\) trials is (Here, \(k<1,000\) ) (A) \(\frac{k}{100}\) (B) \(\frac{k}{1000}\) (C) \(\frac{1,000-k}{1,000}\) (D) none of these

The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that second can be subtracted first one without borrowing is (A) \(\left(\frac{9}{20}\right)^{6}\). (B) \(\frac{1}{2^{6}}\) (C) \(\left(\frac{11}{20}\right)^{6}\) (D) none of these

\(10 \%\) of a certain population suffer from a serious disease. A person suspected of the disease is given two independent tests. Each test makes a correct diagnosis \(90 \%\) of the time. The probability that the person really has the illness given that both tests are positive is (A) \(0.5\) (B) \(0.9\) (C) \(0.6\) (D) none of these

The probablity that the birthdays of six different people will fall in exactly two calendar months is (A) \(\frac{1}{6}\) (B) \({ }^{12} C_{2} \times \frac{2^{6}}{12^{6}}\) (C) \({ }^{12} C_{2} \times \frac{2^{6}-2}{12^{6}}\) (D) \(\frac{341}{12^{5}}\)

A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters \(\mathrm{ON}\) are legible. The probability that it came from LONDON is (A) \(\frac{5}{17}\) (B) \(\frac{12}{17}\) (C) \(\frac{17}{30}\) (D) \(\frac{3}{5}\)

A box contains \(n\) pairs of shoes and \(2 r\) shoes are selected. ( \(r

5 girls and 10 boys sit at random in a row having 15 chairs numbered as 1 to \(15 .\) The probability that end seats are occupied by the girls and between any two girls odd number of boys sit is (A) \(\frac{20 \times 10 ! \times 5 !}{15 !}\) (B) \(\frac{20 \times 10 !}{15 !}\) (C) \(\frac{20 \times 5 !}{15 !}\) (D) none of these

Four tickets marked \(00,01,10,11\), respectively are placed in a bag. A ticket is drawn at random five times, being replaced each time. The probability that the sum of the numbers on tickets thus drawn is 23 is (A) \(\frac{25}{256}\) (B) \(\frac{100}{256}\) (C) \(\frac{231}{256}\) (D) none of these

If \(p\) and \(q\) are chosen randomly from the set \((1,2,3,4,\), \(5,6,7,8,9,10\) ) with replacement then the probability that the roots of the equation \(x^{2}+p x+q=0\) are real, is (A) \(0.62\) (B) \(0.32\) (C) \(0.44\) (D) none of these

Let \(A, B, C\) be three events. If the probability of occurring exactly one event out of \(A\) and \(B\) is \(1-a\), out of \(B\) and \(C\) is \(1-2 a\), out of \(C\) and \(A\) is \(1-a\) and that of occurring three events simultaneously is \(a^{2}\), then the probability that at least one out of \(A, B, C\) will occur, is \((\mathrm{A})<\frac{1}{2}\) (B) \(>\frac{1}{3}\) \((\mathrm{C})>\frac{1}{2}\) (D) \(<\frac{1}{3}\)

Fifteen persons, among whom are \(A\) and \(B\), sit down at random at a round table. The probability that there are 4 persons between \(A\) and \(B\) is (A) \(\frac{1}{3}\) (B) \(\frac{1}{7}\) (C) \(\frac{1}{5}\) (D) none of these

A bag contains \(n\) white and \(n\) red balls. Pairs of balls are drawn without replcement until the bag is empty. The probability that each pair consists of one white and one red ball is (A) \(\frac{2^{n-1}}{{ }^{2 n} C_{n}}\) (B) \(\frac{2^{n-1}}{{ }^{2 n} C_{n-1}}\) (C) \(\frac{2^{n}}{{ }^{2 n} C_{n}}\) (D) none of these

A bag contains \(a\) white and \(b\) black balls. Two players \(A\) and \(B\) alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and and wins the game. If \(A\) begins the game and the probability of \(A\) winning the game is three times that of \(B\), then \(a: b=\) (A) \(2: 1\) (B) \(3: 1\) (C) \(3: 2\) (D) none of these

If \(X\) and \(Y\) are independent binomial variates \(B\left(5, \frac{1}{2}\right)\) and \(B\left(7, \frac{1}{2}\right)\) then the value of \(P(X+Y=3)\) is (A) \(\frac{55}{1024}\) (B) \(\frac{44}{1024}\) (C) \(\frac{33}{1024}\) (D) none of these

Plant I of \(X Y Z\) manufacturing organization employs 5 production and 3 maintenance foremen, another plant II of same organization employs 4 production and 5 maintenance foremen. From any one of these plants, a single selection of two foremen is made. The probability that one of them would be production and the other maintenance foreman is (A) \(\frac{275}{504}\) (B) \(\frac{263}{504}\) (C) \(\frac{301}{504}\) (D) \(\frac{362}{504}\)

In a certain recruitment test there are multiple choice questions. There are 4 possible answers to each question and of which one is correct. An intelligent student knows \(90 \%\) of the answer while a weak student knows only \(20 \%\). If an intelligent student gets the correct answer, then the probability that he was guessing is (A) \(\frac{1}{37}\) (B) \(\frac{36}{37}\) (C) \(\frac{14}{37}\) (D) none of these

In a certain recruitment test there are multiple choice questions. There are 4 possible answers to each question and of which one is correct. An intelligent student knows \(90 \%\) of the answer while a weak student knows only \(20 \%\). If an intelligent student gets the correct answer, then the probability that he was guessing is (A) \(\frac{1}{37}\) (B) \(\frac{36}{37}\) (C) \(\frac{14}{37}\) (D) none of these

If four whole numbers taken at random are multiplied together, the chance that the last digit in the product is \(1,3,7\), or 9 is (A) \(\frac{4}{625}\) (B) \(\frac{18}{625}\) (C) \(\frac{16}{625}\) (D) none of these

A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined, are all red, the probability that the missing card is black is (A) \(\frac{2}{3}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{9}\) (D) none of these

A ten-digit number is formed using the digits from zero to nine, every digit being used exactly once. The probability that the number is divisible by 4 is (A) \(\frac{16}{81}\) (B) \(\frac{20}{81}\) (C) \(\frac{32}{81}\) (D) none of these

Each coefficient of the equation \(a x^{2}+b x+c=0\) is determined by throwing an ordinary die. The probability that the equation has non-real complex roots is (A) \(\frac{173}{216}\) (B) \(\frac{43}{216}\) (C) \(\frac{54}{216}\) (D) none of these

A set \(A\) contains \(n\) elements. \(A\) subset \(P\) of \(A\) is chosen at random and the set \(A\) is reconstructed by replacing the elements of \(P\). Another subset \(\mathrm{Q}\) of \(A\) is now chosen at random. The probability that \(P U Q\) contains exactly \(r\) elements, \(1 \leq r \leq n\) is (A) \(\frac{{ }^{n} C_{r} 3^{r}}{4^{n}}\) (B) \(\frac{{ }^{n} C_{r} 4^{r}}{3^{n}}\)(C) \(\frac{3^{n}}{4^{n}}\) (D) none of these

A person throws two dice, one the common cube and the other a regular tetrahedron, the number on the lowest face being taken in the case of tetrahedron. The probability that the sum of the numbers appearing on the dice is 6 is (A) \(\frac{1}{3}\) (B) \(\frac{1}{4}\) (C) \(\frac{1}{6}\) (D) none of these

\(A\) and \(B\) play a game of tennis. The situation of the game is as follows: if one scores two consecutive points after a deuce, he wins. If loss of a point is followed by win of a point, it is deuce. The probability of a server to win a point is \(\frac{2}{3}\). The game is at deuce and \(A\) is serving. Probability that \(A\) will win the match is (serves are changed after each game ) (A) \(\frac{1}{4}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{2}\) (D) none of these

Let \(0

A bag contains four tickets with numbers 112,121 , \(211,222 .\) One ticket is drawn at random from the bag. Let \(\mathrm{E}_{j}(i=1,2,3)\) denote the event that \(i\) th digit on the drawn ticket is \(2 .\) Then, (A) \(E_{1}, E_{2}, E_{3}\) are pair-wise independent (B) \(E_{1}, \bar{E}_{2}\) are independent (C) \(\bar{E}_{2}\) and \(\bar{E}_{3}\) are not independent (D) \(E_{1}, E_{2}, E_{3}\) are mutually independent

\(A\) and \(B\) are two events. Odds against \(A\) are \(2: 1\). Odds in favour of \(A \cup B\) are \(3: 1\). If \(x \leq P(B) \leq y\), then (A) \(x=\frac{5}{12}\) (B) \(x=\frac{3}{4}\) (C) \(y=\frac{5}{12}\) (D) \(y=\frac{3}{4}\)

If \(A\) and \(B\) are two events such that \(P(A \cup B) \geq \frac{3}{4}\) and \(\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}\), then (A) \(P(A)+P(B) \leq \frac{11}{8}\) (B) \(P(A) \cdot P(B) \leq \frac{3}{8}\) (C) \(P(A)+P(B) \geq \underline{7}\) (D) none of these

A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in tests I, II and III are \(p, q\) and \(\frac{1}{2}\), respectively. If the probability that the student is successful is \(\frac{1}{2}\), then (A) \(p=1\) (B) \(p=0\) (C) \(q=1\) (D) \(q=0\)

A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in tests I, II and III are \(p, q\) and \(\frac{1}{2}\), respectively. If the probability that the student is successful is \(\frac{1}{2}\), then (A) \(p=1\) (B) \(p=0\) (C) \(q=1\) (D) \(q=0\)

The probability that a student passes in Mathematics, Physics and Chemistry are \(m, p\) and \(c\), respectively. Of these subjects, the student has a \(75 \%\) chance of passing in at least one, a \(50 \%\) chance of passing in at least two and a \(40 \%\) chance of passing in exactly two. Which of the following relations are true? (A) \(p+m+c=19 / 20\) (B) \(p+m+c=27 / 20\) (C) \(p m c=1 / 10\) (D) \(p m c=1 / 4\)

If \(A\) and \(B\) are two events such that \(P(A)=\frac{1}{2}\) and \(P(B)=\frac{2}{3}\), then (A) \(P(A \cup B) \geq \frac{2}{3}\) (B) \(P\left(A \cap B^{\prime}\right) \leq \frac{1}{3}\) (C) \(1 / 6 \leq P(A \cap B) \leq \frac{1}{2}\) (D) \(1 / 6 \leq P\left(A^{\prime} \cap B\right) \leq \frac{1}{2}\)

If \(P(A)=\frac{2}{5}\) and \(P(B)=\frac{4}{5}\), then (A) \(P(A \cup B) \geq \frac{4}{5}\) (B) \(\frac{1}{5} \leq P(A \cap B) \leq \frac{2}{5}\) (C) \(\frac{1}{4} \leq P(A / B) \leq \frac{1}{2}\) (D) \(P\left(A \cap B^{\prime}\right) \leq \frac{1}{5}\)

If \(A\) and \(B\) are two events, then the probability that at most one of \(A, B\) occurs is (A) \(1-P(A \cap B)\) (B) \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)-P\left(A^{\prime} \cap B^{\prime}\right)\) (C) \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)+P(A \cup B)\) (D) none of these