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

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

In an entrance test there are multiple choice questions. There are four possible answers to each question of which one is correct. The probability that a student knows the answer to a question is \(90 \%\). If he gets the correct answer to a question, then the probability that he was guessing is (A) \(\frac{1}{9}\) (B) \(\frac{36}{37}\) (C) \(\frac{1}{37}\) (D) \(\frac{47}{40}\)

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) \(\frac{1}{64}\)

\(A\) 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 it 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) \(\frac{19}{101}\)

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 \(\mathrm{S}_{1}\) is among the eight winners is (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) none of these

Three numbers are chosen at random without replcement 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

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) \(\frac{1}{4}\)

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) \(\frac{1}{4}\)

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 thenumber 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) \(\frac{65}{81}\)

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) \(\frac{1}{75}\)

The probabilities of four cricketers \(A, B, C\) and \(D\) scoring more than 50 runs in a match are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\) and \(\frac{1}{10} .\) It is known that exactly two of the players scored more than 50 runs in a particular match. The probability that these players were \(A\) and \(B\) is (A) \(\frac{27}{65}\) (B) \(\frac{5}{6}\) (C) \(\frac{1}{6}\) (D) none of these

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

If \(a \in[-5,30]\), then the probability that the graph of the function \(y=x^{2}+2(a+4) x-5 a+64\) is strictly above the \(x\)-axis is (A) \(\frac{27}{35}\) (B) \(\frac{8}{25}\) (C) \(\frac{8}{35}\) (D) \(\frac{17}{25}\)

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

vSuppose \(n(\geq 3)\) persons are sitting in a row. Two of them are selected at random. The probability that they are not together is (A) \(1-\frac{2}{n}\) (B) \(\frac{2}{n-1}\) (C) \(1-\frac{1}{n}\) (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

\(2 n\) 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

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

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

An ordinary cube has four blank faces, one face marked 2 and another marked 3. Then the probability of obtaining 9 in 5 throws is(A) \(\frac{31}{7776}\) (B) \(\frac{5}{2592}\) (C) \(\frac{5}{1944}\) (D) \(\frac{5}{1296}\)

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

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

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) \(\frac{65}{81}\)

In a game "odd man out" each of \(m \geq 2\) persons, tosses a coin to determine who will buy refreshments for the entire group. The odd man out is the one with a different outcome from the rest. The probability that there is a loser in any game is (A) \(\frac{1}{2^{m-1}}\) (B) \(\frac{m-1}{2^{m-1}}\) (C) \(\frac{m}{2^{m-1}}\) (D) none of these

If \(A_{1}, A_{2}, \ldots, A_{n}\) are \(n\) independent events such that \(P(A)\) \(=\frac{1}{i+1}, i=1,2, \ldots, n\). The probability that none of the \(n\) events occurs is (A) \(\frac{n}{n+1}\) (B) \(\frac{1}{n+1}\) (C) \(\frac{n}{(n+1)(n+2)}\) (D) none of these

Consider 5 independent Bernoulli's trials each with probability of success \(p\). If the probability of at least one failure is greater than or equal to \(31 / 32\), then \(p\) lies in the interval (A) \(\left(\frac{11}{12}, 1\right)\) (B) \(\left(\frac{1}{2}, \frac{3}{4}\right]\) (C) \(\left(\frac{3}{4}, \frac{11}{12}\right]\) (D) \(\left[0, \frac{1}{2}\right]\)

If \(C\) and \(D\) are two events such that \(C \subset D\) and \(P(D) \neq\) 0 , then the correct statement among the following is (A) \(P(C \mid D)=\frac{P(D)}{P(C)}\) (B) \(P(C \mid D)=P(C)\) (C) \(P(C \mid D) \geq P(C)\) (D) \(P(C \mid D)