Chapter 24

One hundred identical coins, each with probability \(p\) of showing up heads, are tossed. If \(0

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) \(P(A)+P(B) \geq \frac{7}{8}\) (D) none of these

A point is selected at random from the interior of a circle. The probability that the point is closer to the centre than the boundary of the circle is (A) \(\frac{3}{4}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) none of these

From a box containing 20 tickets of value 1 to 20 , four tickets are drawn one by one. After each draw, the ticket is replaced. The proability that the largest value of tickets drawn is 15 is (A) \(\left(\frac{3}{4}\right)^{4}\) (B) \(\frac{27}{320}\) (C) \(\frac{27}{1280}\) (D) none of these

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

A person draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues doing this until be draws a spade, the chance that he will fail in the first two draws is (A) \(\frac{1}{16}\) (B) \(\frac{9}{16}\) (C) \(\frac{9}{64}\) (D) \(\frac{1}{64}\)

For two events \(A\) and \(B\) if \(P(A)=P\left(\frac{A}{B}\right)=\frac{1}{4}\) and \(P\left(\frac{B}{A}\right)=\frac{1}{2}\), then (A) \(A\) is subevent of \(B\) (B) \(A\) and \(B\) are mutually exclusive (C) \(A\) and \(B\) are independent and \(P\left(\frac{A^{\prime}}{B}\right)=\frac{3}{4}\) (D) none of the above

\(A\) and \(B\) throw a dice. The probability that \(A\) 's throw is not greater than B's is (A) \(\frac{5}{12}\) (B) \(\frac{7}{12}\) (C) \(\frac{1}{6}\) (D) \(\frac{1}{2}\)

A six faced die is so biased that is twice likely to show an even number as compared to an odd number when thrown. The die is thorwn twice. The probability that the sum of the two numbers is even is (A) \(\frac{5}{9}\) (B) \(\frac{4}{9}\) (C) \(\frac{1}{3}\) (D) none of these

\(n\) biscuits are distributed among \(N\) boys at random. The probability that particular boy gets \(r(

Cards are drawn from a pack of 52 cards one by one. The probability that exactly 10 cards will be drawn before the first ace is (A) \(\frac{451}{884}\) (B) \(\frac{241}{1456}\) (C) \(\frac{164}{4165}\) (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) \(p=1, q=\frac{1}{2}\) (B) \(p=1, q=0\) (C) \(p=q=1 \frac{1}{2}\) (D) \(p=q=1\).

A sum of money is rounded off to the nearest rupee. The probability that round off error is at least ten paise is (A) \(\frac{81}{100}\) (B) \(\frac{82}{101}\) (C) \(\frac{19}{100}\) (D) \(\frac{19}{101}\)

A sum of money is rounded off to the nearest rupee. The probability that round off error is at least ten paise is (A) \(\frac{81}{100}\) (B) \(\frac{82}{101}\) (C) \(\frac{19}{100}\) (D) \(\frac{19}{101}\)

Sixteen players \(S_{1}, S_{2}, \ldots S_{16}\) play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the palyers are to equal strength. The probability that the players \(S_{1}\) is among the eight winners is (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) none of these

There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identifed. Then the probability that only two tests are needed is (A) \(\frac{1}{3}\) (B) \(\frac{1}{6}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{4}\)

If two events \(A\) and \(B\) are such that \(P\left(A^{2}\right)=0.3, P(B)\) \(=0.4\) and \(P\left(A B^{\circ}\right)=0.5\), then \(P\left[B \backslash\left(A \cup B^{c}\right)\right]=\) (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{4}\) (D) none of these

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shufled pack of eleven cards numbered \(2,3,4, \ldots, 12\) is picked and the number on the card is noted. The probability that the noted number is either 7 or 8 , is (A) \(\frac{193}{792}\) (B) \(\frac{164}{792}\) (C) \(\frac{231}{792}\) (D) none of these

An unbiased die with faces marked \(1,2,3,4,5\) and 6 is rolled four times. Out of four face values obtained, the probabiity that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is, (A) \(\frac{16}{81}\) (B) \(\frac{1}{81}\) (C) \(\frac{80}{81}-\) (D) \(\frac{65}{81}\)

Numbers are selected at random, one at a time, from the two-digit numbers \(00,01,02, \ldots, 99\) with replacement. An even E occurs if and only if the product of the two digits of selected number is 18 . If four numbers are selected, the probability that the event E occurs at least 3 times, is (A) \(\frac{99}{(25)^{4}}\) (B) \(\frac{86}{(25)^{4}}\) (C) \(\frac{74}{(25)^{4}}\) (D) \(\frac{97}{(25)^{4}}\)

These are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is (A) \(\frac{5}{8}\) (B) \(\frac{3}{8}\) (C) \(\frac{1}{8}\) (D) none of these

Two small squares on a chess board are chosen at random. Probability that they have a common side is (A) \(1 / 3\) (B) \(1 / 9\) (C) \(1 / 18\) (D) none of these

Three winning tickets are drawn from an urn of 100 tickets. The probability of winning for a person who buys 4 tickets is (A) \(\frac{7144}{8085}\) (B) \(\frac{941}{8085}\) (C) \(\frac{6321}{8085}\) (D) none of these

A five digit number is selected at random. Then the probability that the digits in the odd places are odd and in the even places are even (no digit being repeated) is (A) \(\frac{9}{10}\) (B) \(\frac{1}{10}\) (C) \(\frac{1}{25}\) (D) \(\frac{1}{75}\)

A five digit number is selected at random. Then the probability that the digits in the odd places are odd and in the even places are even (no digit being repeated) is (A) \(\frac{9}{10}\) (B) \(\frac{1}{10}\) (C) \(\frac{1}{25}\) (D) \(\frac{1}{75}\)

The numbers \(1,2,3, \ldots, n\) are arranged in a random order. The probability that the digits \(1,2,3, \ldots k\) ( \(k

Fifteen coupons are numbered \(1,2,3, \ldots 15 .\) Seven coupons are selected at random one at a time withreplacement. 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

A bag contains \(m\) white and 3 black balls. Balls are drawn one by one without replacement till all the black balls are drawn. The probability that this procedure for drawing balls will come to an end at the \(r\) th draw is (A) \(\frac{(r-1)(r-2)}{(m+1)(m+2)(m+3)}\) (B) \(\frac{3(r-1)(r-2)}{(m+1)(m+2)(m+3)}\) (C) \(\frac{2(r-1)(r-2)}{(m+1)(m+2)(m+3)}\) (D) none of these

A fair die is tossed eight times. Probability that on the eighth throw a third six is observed is (A) \({ }^{8} C_{3} \frac{5^{5}}{6^{8}}\) (B) \(\frac{{ }^{7} C_{2} \cdot 5^{5}}{6^{8}}\) (C) \(\frac{{ }^{7} C_{2} \cdot 5^{5}}{6^{7}}\) (D) none of these

A natural number \(x\) is chosen at random from the first one hundred natural numbers. The probability that \(\frac{(x-20)(x-40)}{x-30}<0\) is (A) \(\frac{1}{50}\) (B) \(\frac{9}{50}\) (C) \(\frac{3}{25}\) (D) \(\frac{7}{25}\)

Four five-rupee coins, 3 two-rupee coins and 2 one-rupee coins are stacked together in a column at random. The probability that the coins of the same denomination are consecutive is (A) \(\frac{13}{9 !}\) (B) \(\frac{1}{210}\) (C) \(\frac{1}{35}\) (D) none of these

Numbers \(1,2,3, \ldots, 2 n(n \in N)\) are printed on \(2 n\) cards. The probability of drawing a number \(n\) is proportional to \(r\). Then the probability of drawing an even number in one draw is (A) \(\frac{n+2}{n+3}\) (B) \(\frac{n+1}{n+3}\) (C) \(\frac{1}{2}\) (D) \(\frac{n+1}{2 n+1}\)

Three natural numbers are taken at random from the set \(A=\\{x: 1 \leq x \leq 100, x \in N\\} .\) The probabiulity that the \(A . M\). of the numbers taken is 25 is (A) \(\frac{{ }^{77} C_{2}}{{ }^{100} C_{3}}\) (B) \(\frac{{ }^{25} C_{2}}{{ }^{100} C_{3}}\) (C) \(\frac{{ }^{74} C_{2}}{{ }^{100} C_{3}}\) (D) none of these

\(2 n\) boys are randomly divided into two subgroups containing \(n\) boys each. The probability that the two tallest boys are in different groups is (A) \(\frac{n}{2 n-1}\) (B) \(\frac{n-1}{2 n-1}\) (C) \(\frac{2 n-1}{4 n^{2}}\) (D) none of these

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

A car is parked by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 places are still occupied. The probability that both theneighbouring places are empty is (A) \(\frac{91}{276}\) (B) \(\frac{15}{184}\) (C) \(\frac{15}{92}\) (D) none of these

In a corner hexagon two diagonals are drawn at random. The probability that diagonals intersect at an interior point of the hexagon is (A) \(\frac{5}{12}\) (B) \(\frac{7}{12}\) (C) \(\frac{2}{5}\) (D) none of these

Six different balls are put in three different boxes, no box being empty. The probability of putting balls in the boxes in equal numbers is, (A) \(3 / 10\) (B) \(1 / 6\) (C) \(1 / 5\) (D) none of these

At a railway staition a passenger leaves his luggage in a locker which is opened by dialling a three digit code (say \(253,009,325\) etc.). 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 \(k<1000\) ) (A) \(\frac{k}{100}\) (B) \(\frac{k}{1000}\) (C) \(\frac{1000 \quad k}{1000}\) (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

There are \(n\) persons \((n \geq 3)\), among whom are \(A\) and \(B\), who are made to stand in a row in random order. Probability that there is exactly one person between \(A\) and \(B\) is (A) \(\frac{n-2}{n(n-1)}\) (B) \(\frac{2(n-2)}{n(n-1)}\) (C) \(2 / n\) (D) none os these

\(x_{1}, x_{2}, x_{3}, \ldots x_{50}\) are fifty real numbers such that \(x_{r}

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) \({ }^{12} C_{2} \times \frac{2^{6}-1}{12^{6}}\) (D) \(\frac{341}{12^{5}}\)

A bag contains \((2 n+1)\) coins. It is known that \(n\) of these coins have a head on both sides, whereas the remaining \(n+1\) coins are fair. \(A\) coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is \(\frac{31}{42}\), then \(n\) is equal to (A) 10 (B) 11 (C) 12 (D) 13

One mapping is selected at random from all mappings of the set \(S=\\{1,2,3, \ldots n\\}\) into itself. The probability that it is one one is \(\frac{3}{32}\). Then the value of \(n\) is (A) 3 (B) 4 (C) 5 (D) 6

Suppose \(n\) people are asked a question successively in a random order and exactly 3 of the \(n\) people know that answer. If \(n>6\), the probability that the first four of those asked do not know the answer is (A) \(\frac{{ }^{n-4} C_{4}}{{ }^{n} C_{4}}\) (B) \(\frac{{ }^{n-3} C_{4}}{{ }^{n} C_{4}}\) (C) \(\frac{1}{{ }^{n} C_{4}}\) (D) none of these

A digit is selected from each of the following two sets: \(I=\\{0,1,2,3,4,5,6,7,8,9\\}\) \(I I=\\{0,1,2,3,4,5,6,7,8,9\\}\) The probability that the product of the digits so chosen is positive is (A) \(\frac{4}{5}\) (B) \(\frac{81}{100}\) (C) \(\frac{91}{100}\) (D) none of these

A square is inscribed in a circle. If \(p_{1}\) is the probability that a randomly chosen point of the circle lies within the square and \(p_{2}\) is the probability that the point lies outside the square then(A) \(p_{1}=p_{2}\) (B) \(p_{1}>p_{2}\) and \(p_{1}^{2}-p_{2}^{2}<\frac{1}{3}\) (C) \(p_{1}

An unbiased die with faces marked \(1,2,3,4,5\) and 6 is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is (A) \(\frac{16}{81}\) (B) \(\frac{1}{81}\) (C) \(\frac{80}{81}\) (D) \(\frac{65}{81}\)