Chapter 3

Seven chits are numbered 1 to 7 . Three are drawn one by one with replacement. The probability that the least number on any selected chit is 5 is (a) \(1-\left(\frac{2}{7}\right)^{4}\) (b) \(4\left(\frac{2}{7}\right)^{4}\) (c) \(\left(\frac{3}{7}\right)^{3}\) (d) None of these

A group of children contains 6 boys and 4 girls. Three children are chosen at random from this group. Find the probability that this group chosen: (i) contains only a particular girl. (ii) contains at least one girl.

Adice is thrown twice and the sum of the numbers appearing is observed to be 6 . What is the conditional probability that the number 4 has appeared at least once?

If \(4 P(A)=6 P(B)=10(A \cap B)=1\), then \(P(B / \mathrm{A})=\) (a) \(2 / 5\) (b) \(3 / 5\) (c) \(7 / 10\) (d) \(19 / 60\)

A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and kept aside. From the remaining balls in the box, another ball is drawn at random and kept aside the first. This process is repeated till all the balls are drawn from the box. The probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red is (a) \(\frac{1}{1260}\) (b) \(\frac{1}{7560}\) (c) \(\frac{1}{126}\) (d) None of these

A speaks untruth in \(30 \%\) cases and \(B\) speaks truth in \(60 \%\) cases. Find the probability when they contradict each other.

A player draws a playing card from a set of playing cards. What will be the probability of not being a diamond card? \([\mathrm{MP}-2004(\mathrm{~A})]\)

A fair coin is tossed \(n\) times. If the probability that head occurs six times is equal to the probability that head occurs eight times, then \(n\) is equal to (a) 15 (b) 14 (c) 12 (d) 7

Three coins are thrown simultaneously. Find: (i) probability of getting at least two heads. (ii) probability of getting at most two heads.

The probability that a teacher will give an unannounced test during any class meeting is \(1 / 5\). If a student is absent twice, then the probability that the student will miss at least one test is (a) \(4 / 5\) (b) \(2 / 5\) (c) \(7 / 5\) (d) \(9 / 25\)

In three groups of children there are 3 girls and 1 boy, 2 girls and 2 boys, 1 girl and 3 boys, respectively. One child is chosen at random from each group. Prove that if there is 1 girl and 2 boys among the chosen children then the probability is \(13 / 32\) [MP-99; Roorkee-85; CBSE-Sample Paper-IIIJ

A question of mathematics is given to three students to solve. Probabilities of solving the question by them are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\), respectively. If they try to solve it, what is the probability that the problem will be solved?

A bag contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If one item is taken at random. Find the probability that it is rusted or is a bolt. [MP-2000]

If the probability that a student is not a swimmer is \(1 / 5\), then the probability that out of 5 students 1 is swimmer is (a) \({ }^{5} C_{1}\left(\frac{4}{5}\right)^{4}\left(\frac{1}{5}\right)\) (b) \({ }^{5} C_{1} \frac{4}{5}\left(\frac{1}{5}\right)^{4}\) (c) \(\frac{4}{5}\left(\frac{1}{5}\right)^{4}\) (d) None of these

Two dice are thrown simultaneously. Find the probability that the first die shows an even number or both the dice show the sum 8 .

Mohan tells the truth in \(75 \%\) cases while Sohan in \(80 \%\) cases. Find the probability that Mohan tells the truth and Sohan tells lie to narrate an incident. [MP-2001]

A bag contains 3 red and 7 black balls, 2 balls are taken out at random, without replacement. If the first ball taken out is red, then what is the probability that the second taken out ball is also red? (a) \(1 / 10\) (b) \(1 / 15\) (c) \(3 / 10\) (d) \(2 / 21\)

A coin is tossed \(n\) times. The probability of getting head at least once is greater than \(0.8\), then the least value of \(n\) is (a) 2 (b) 3 (c) 4 (d) 5

If \(P(A)=\frac{1}{2}, P(B)=\frac{1}{4}\) and \(P(A \cap B)=\frac{1}{4}\), then find the value of the following: (i) \(P\left(\frac{A}{B}\right)\) (ii) \(P\left(\frac{B}{A}\right)\) (iii) \(P(A \cup B)\)

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find the probability distribution of the number of jacks. [CBSE-2006 (outside-Delhi)-I, II and III]

A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. A ball is picked up randomly. The probability of its being black is (a) \(2 / 5\) (b) \(8 / 15\) (c) \(6 / 11\) (d) \(2 / 3\)

A die is tossed five times. Getting an odd number is considered a success. Then the variance of distribution of the success is (a) \(8 / 3\) (b) \(3 / 8\) (c) \(4 / 5\) (d) \(5 / 4\)

\(A\) can solve \(90 \%\) of the problem given in a book and \(B\) can solve \(70 \%\). What is the prob-ability that at least one of them will solve a problem selected at random from the book. [MP-99, 2003; CBSE-92(C)]

The probability of student \(A\) passing an examination is \(3 / 7\) and of student \(B\) passing is \(5 / 7\). Assuming the two events ' \(A\) passes', ' \(B\) passes', as independent, find the probability of: (i) only a passing the examination. (ii) only one of them passing the examination.

A box containing 4 white and 2 black pens. Another box contains 3 white and 5 black pens. If 1 pen is selected from each box, then the probability that both the pens are white is equal to (a) \(\frac{1}{2}\) (b) \(\frac{1}{3}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{5}\)

The mean and variance of a binomial distribution are 4 and 3, respectively, then the probability of getting exactly 6 successes in this distribution is (a) \({ }^{16} C_{6}\left(\frac{1}{4}\right)^{10}\left(\frac{3}{4}\right)^{6}\) (b) \({ }^{16} C_{6}\left(\frac{1}{4}\right)^{6}\left(\frac{3}{4}\right)^{10}\) (c) \({ }^{12} C_{6}\left(\frac{1}{4}\right)^{10}\left(\frac{3}{4}\right)^{6}\) (d) \({ }^{12} C_{6}\left(\frac{1}{4}\right)^{6}\left(\frac{3}{4}\right)^{6}\)

Out of 9 outstanding students in a college, there are 4 boys and 5 girls. A team of four students is to be selected for a quiz programme. Find the probability that 2 are boys and 2 are girls. [CBSE-94]

A box contains 25 tickets numbered \(1,2, \ldots\) 25. If 2 tickets are drawn at random, then the probability that the product of their numbers is even is (a) \(11 / 50\) (b) \(13 / 50\) (c) \(37 / 50\) (d) None of these

In a college \(25 \%\) students fail in maths, \(15 \%\) fail in chemistry and \(10 \%\) students fail in maths and chemistry both. A student is selected at random, then (i) What is the probability that he fails in Maths, if he is failed in Chemistry? (ii) What is the probability that he fails in Chemistry, if he is failed in Maths? (iii) What is the probability that he is failed in Maths or Chemistry?

A fair die is tossed twice. If the number appearing on the top is less than 3 , it is a success. Find the probability distribution of the number of successes. [CBSE-2004]

A box contains 16 bulbs; out of which 4 bulbs are defective. Three bulbs are drawn one by one from the box without replacement. Find the probability distribution of the number of defective bulbs drawn.

A box contains 10 mangoes; out of which 4 are rotten. Two mangoes are taken out together. If one of them is found to be good, the probability that the other is also good is (a) \(1 / 3\) (b) \(8 / 15\) (c) \(5 / 13\) (d) \(2 / 3\)

In a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4 . Find the probability that it is actually a number greater than 4 . [CBSE-2009]

A party of 23 persons take their seats at a round table. The odds against two persons sitting together are (a) \(10: 1\) (b) \(1: 11\) (c) \(9: 10\) (d) None of these

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

Coloured balls are distributed in three bags, as shown in the following table: $$ \begin{array}{lccc} \text { Bag } & \text { Black } & \text { White } & \text { Red } \\ \text { I } & 2 & 1 & 3 \\ \text { II } & 4 & 2 & 1 \\ \text { III } & 5 & 4 & 3 \end{array} $$ A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be white and red. What is the probability that they came from bag II?

If \(E\) and \(F\) are independent events such that 0 \(

Seven white balls and 3 black balls are randomly placed in a row. The probability that no 2 black balls are placed adjacently equals (a) \(1 / 2\) (b) \(7 / 15\) (c) \(2 / 15\) (d) \(1 / 3\)

If two events \(A\) and \(B\) are such that \(P\left(A^{c}\right)=0.3\), \(P(B)=0.4\) and \(P\left(A B^{c}\right)=0.5\), then \(P\left[B /\left(A \cup B^{c}\right)\right]\) is equal to (a) \(1 / 2\) (b) \(1 / 3\) (c) \(1 / 4\) (d) None of these

A bag contains 30 balls numbered from 1 to 30; 1 ball is drawn randomly. The probability that the number of the ball is multiple of 5 or 7 is (a) \(1 / 2\) (b) \(1 / 3\) (c) \(2 / 3\) (d) \(1 / 4\)

Two cards are drawn one by one from a pack of cards. The probability of getting first card an ace and second a coloured one is (before drawing the second card the first card is not placed again in the pack) (a) \(\frac{1}{26}\) (b) \(\frac{5}{52}\) (c) \(\frac{5}{221}\) (d) \(\frac{4}{13}\)

The probabilities of a student getting I, II and III division in an examination are \(1 / 10,3 / 5\) and \(1 / 4\), respectively. The probability that the student fails in the examination is (a) \(197 / 200\) (b) \(27 / 100\) (c) \(83 / 100\) (d) None of these

A bag \(X\) contains 2 white and 3 black balls and another bag \(Y\) contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen be white is (a) \(\frac{2}{15}\) (b) \(\frac{7}{15}\) (c) \(\frac{8}{15}\) (d) \(\frac{14}{15}\)

In tossing 10 coins, the probability of getting exactly 5 heads is (a) \(9 / 128\) (b) \(63 / 256\) (c) \(1 / 2\) (d) \(193 / 256\)

In a box containing 100 eggs, 10 eggs are rotten. The probability that out of a sample of 5 eggs none is rotten if the sampling is with replacement is (a) \(\left(\frac{1}{10}\right)^{5}\) (b) \(\left(\frac{1}{5}\right)^{5}\) (c) \(\left(\frac{9}{5}\right)^{5}\) (d) \(\left(\frac{9}{10}\right)^{5}\)

A coin is tossed three times in succession. If \(E\) is the event that there are at least 2 heads and \(F\) is the event in which the first throw is a head, then \(P(E / F)=\) (a) \(3 / 4\) (b) \(3 / 8\) (c) \(1 / 2\) (d) \(1 / 8\)

If a party of \(n\) persons sits at a round table then the odds against two specified individuals sitting next to each other are (a) \(2:(n-3)\) (b) \((n-3): 2\) (c) \((n-2): 2\) (d) \(2:(n-2)\)

A bag contains 2 white and 4 black balls. A ball is drawn five times with replacement. The probability that at least 4 of the balls drawn are white is (a) \(\frac{8}{141}\) (b) \(\frac{10}{243}\) (c) \(\frac{11}{243}\) (d) \(\frac{8}{41}\)

The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chances of the events are (a) \(1 / 9,1 / 3\) (b) \(1 / 16,1 / 4\) (c) \(1 / 4,1 / 2\) (d) None of these