# Ch13. Probability

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## Certificate

Certificate on successful completion of this course.

In this online course, you will learn conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.

## Course Content

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Probability Distribution
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Theorem of Total Probability and Bayes Theorem
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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

Question 1. A coin is tossed and a die is thrown, find the probability of getting odd number.
Question 2. A coin is tossed and then a die is rolled only in case a head is shown on the coin, find the probability of getting odd number.
Question 3. 2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y. Find the probability of selecting a girl.
Question 4. An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the probability of getting two heads.
Question 5. Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non – defective(N). Write the sample space of this experiment and find the probability of getting at least two defective bulbs.
Question 6. A coin is tossed. If the outcome is a head, a die is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for the experiment? Find the probability of getting a tail.
Question 7. A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment and find the probability of getting a black ball.
Question 8. An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black? {\displaystyle \left( {\frac{3}{7}} \right)}
Question 9. Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace? {\displaystyle \left( {\frac{2}{{5525}}} \right)}
Question 10. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
a. both balls are red.
b. first ball is black and second is red.
c. one of them is black and other is red. {\displaystyle \left( {\frac{{16}}{{81}},\frac{{20}}{{81}},\frac{{40}}{{81}}} \right)}
Question 11. Find the probability of obtaining an even prime number on each die, when a pair of dice is rolled. {\displaystyle \left( {\frac{1}{{36}}} \right)}
Question 12. A family has two children. What is the probability that both the children are boys given that at least one of them is a boy? {\displaystyle \left( {\frac{1}{3}} \right)}
Question 13. Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number? {\displaystyle \left( {\frac{4}{7}} \right)}

### Assignment – 2

Question 1. In a school, there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in Class XII given that the chosen student is a girl? {\displaystyle (0.1)}
Question 2. A die is thrown three times. Events A and B are defined as below:
A : 4 on the third throw B : 6 on the first and 5 on the second throw
Find the probability of A given that B has already occurred. {\displaystyle \left( {\frac{1}{6}} \right)}
Question 3. A die 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? {\displaystyle \left( {\frac{2}{5}} \right)}
Question 4. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 4’ given that ‘there is at least one tail’. {\displaystyle \left( {\frac{2}{9}} \right)}
Question 5. A black and a red dice are rolled.
a. Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. {\displaystyle \left( {\frac{1}{3}} \right)}
b. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. {\displaystyle \left( {\frac{1}{9}} \right)}
Question 6. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl? {\displaystyle \left( {\frac{1}{2},\frac{1}{3}} \right)}
Question 7. An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple-choice question? {\displaystyle \left( {\frac{5}{9}} \right)}
Question 8. Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’. {\displaystyle \left( {\frac{1}{{15}}} \right)}
Question 9. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die 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 die shows a 3’. {\displaystyle (\,0\,)}
Question 10. Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P(E|F) and P(F|E). {\displaystyle \left( {\frac{2}{3},\frac{1}{3}} \right)}
Question 11. Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32. {\displaystyle \left( {\frac{{16}}{{25}}} \right)}
Question 12. If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find {\displaystyle (0.32,\,\,0.64,\,\,0.98)}
i. P(A ∩ B)
ii. P(A|B)
iii. P(A ∪ B)
Question 13. Evaluate P(A ∪ B), if 2P(A) = P(B) = {\displaystyle \frac{5}{{13}}} and P(A|B)= {\displaystyle \frac{2}{5}} . {\displaystyle \left( {\frac{{11}}{{26}}} \right)}
Question 14. If P(A) = {\displaystyle \frac{6}{{11}}} , P(B) = {\displaystyle \frac{5}{{11}}} and P(A ∪ B)= {\displaystyle \frac{7}{{11}}} , find {\displaystyle \left( {\frac{4}{{11}},\frac{4}{5},\frac{2}{3}} \right)}
i. P(A∩B)
ii. P(A|B)
iii. P(B|A)

### Assignment – 3

Question 1. A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.
Question 2. A die is rolled. If the outcome is an even number, what is the probability that it is a prime?
Question 3. In a class of 25 students with roll numbers 1 to 25, a student is picked up at random to answer a question. Find the probability that the roll number of the selected student is either a multiple of 5 or of 7.
Question 4. Two dice are thrown once. Find the probability of getting an even number on the first die or a total of 8.
Question 5. A can hit a target is 4 times out of 5 times. B can hit a target is 3 times out of 4 times and C can hit a target is 2 times out of 3 times. They fire simultaneously. Find the probability that :
a. Any two will hit the target
b. None of them will hit the target
Question 6. The probability that A, B and C will solve a problem are {\displaystyle \frac{1}{3}} , {\displaystyle \frac{2}{7}} and {\displaystyle \frac{3}{8}} respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it.
Question 7. Probabilities of solving a specific problem independently by A and B are {\displaystyle \frac{1}{2}} and {\displaystyle \frac{1}{3}} respectively, if both try to solve the problem independently, find the probability that
a. The problem is solved
b. Exactly one of them will solve the problem
Question 8. The probability of students A passing the examination is {\displaystyle \frac{3}{5}} and of student B is {\displaystyle \frac{4}{5}} . Assuming that the two events “A passes” and “B passes” as independent. Find the probability of :
a. Both students passing the examination
b. Only A passing the examination
c. Only one of them passing the examination
d. None of them passing the examination
Question 9. In a class 40% students study Statistics, 25% studying Mathematics and 15% both Mathematics and Statistics. One student is selected at random. Find the probability
a. That he study Statistics, if it is known that he studies Mathematics.
b. That he study Mathematics, if it is known that he studies Statistics.
Question 10. In a class 35% students are poor , 20% are meritorious and 15% both poor and meritorious. One student is selected at random. Find the probability that
a. He is poor, if it is known that he is meritorious
b. He is meritorious, if he known that he is poor
Question 11. If {\displaystyle P(A) = \frac{3}{8}} , {\displaystyle P(B) = \frac{1}{2}} and {\displaystyle P(A \cap B) = \frac{1}{4}} , find (i) {\displaystyle P(\overline A |\overline B ),} (ii) {\displaystyle P(\overline B |\overline A )}
Question 12. A pair if dice thrown. Find the probability of getting 7 as a sum, if it is known that second dice always exhibits an odd number.
Question 13. A pair if dice thrown. Find the probability of getting 7 as a sum, if it is known that second dice always exhibits a prime number.
Question 14. Find the probability that the sum of the numbers showing on the two dice is 8,given that at least one dice does not show 5.
Question 15. Two dice are thrown. Find the probability that the numbers appeared have a sum 8, if it is known that second dice always exhibits 4.
Question 16. A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of the cases they are likely to contradict each other in stating the same fact.
Question 17. One card is drawn from a well shuffled pack of 52 cards. If E is the event “ the card drawn is a king or a queen” and F is the event “ the card drawn is the ace or queen”, then find the probability of the conditional even {\displaystyle P(E|F)} .
Question 18. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is the girl (ii) at least one is a girl ?
Question 19. A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.
Question 20. What is the probability of 53 Sundays in a (i) Ordinary year (ii) Leap year

### Assignment – 4

Question 1. A die is tossed thrice. Find the probability of getting an odd number at least once. {\displaystyle \left( {\frac{7}{8}} \right)}
Question 2. 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
a. all will bear ‘X’ mark. {\displaystyle {\left( {\frac{2}{5}} \right)^6}}
b. not more than 2 will bear ‘Y’ mark. {\displaystyle \left( {7{{\left( {\frac{2}{5}} \right)}^4}} \right)}
c. at least one ball will bear ‘Y’ mark. {\displaystyle \left( {1 - {{\left( {\frac{2}{5}} \right)}^6}} \right)}
d. the number of balls with ‘X’ mark and ‘Y’ mark will be equal. {\displaystyle \left( {\frac{{864}}{{3125}}} \right)}
Question 3. A die 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? {\displaystyle \left( {\frac{3}{{32}},\frac{7}{{64}},\frac{{63}}{{64}}} \right)}
Question 4. If a fair coin is tossed 10 times, find the probability of
c. at most six heads {\displaystyle \left( {\frac{{105}}{{512}},\frac{{193}}{{512}},\frac{{53}}{{64}}} \right)}
Question 5. Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg. {\displaystyle \left( {1 - \frac{{{9^{10}}}}{{{{10}^{10}}}}} \right)}
Question 6. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes. {\displaystyle \left( {\frac{{25}}{{216}}} \right)}
Question 7. 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? {\displaystyle \left( {\left( {\frac{{29}}{{20}}} \right){{\left( {\frac{{19}}{{20}}} \right)}^9}} \right)}
Question 8. 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? (ii) only 3 cards are spades? (iii) none is a spade? {\displaystyle \left( {\frac{1}{{1024}},\frac{{45}}{{512}},\frac{{243}}{{1024}}} \right)}
Question 9. 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
a. None {\displaystyle ({(0.95)^5})}
b. not more than one {\displaystyle ({(0.95)^4} \times 1.2)}
c. more than one {\displaystyle (1 - {(0.95)^4} \times 1.2)}
d. at least one will fuse after 150 days of use. {\displaystyle (1 - {(0.95)^5})}
Question 10. 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? {\displaystyle {\left( {\frac{9}{{10}}} \right)^4}}
Question 11. In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly. {\displaystyle \left( {{{\left( {\frac{1}{2}} \right)}^{20}}{[^{20}}{C_{12}}{ + ^{20}}{C_{13}} + ...{ + ^{20}}{C_{20}}]} \right)}
Question 12. On 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? {\displaystyle \left( {\frac{{11}}{{243}}} \right)}
Question 13. A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is {\displaystyle \frac{1}{{100}}} . What is the probability that he will win a prize
a. at least once {\displaystyle \left( {1 - {{\left( {\frac{{99}}{{100}}} \right)}^{50}}} \right)}
b. exactly once {\displaystyle \left( {\frac{1}{2}{{\left( {\frac{{99}}{{100}}} \right)}^{49}}} \right)}
c. at least twice? {\displaystyle \left( {1 - \frac{{149}}{{100}}{{\left( {\frac{{99}}{{100}}} \right)}^{49}}} \right)}
Question 14. Find the probability of getting 5 exactly twice in 7 throws of a die. {\displaystyle \left( {\frac{7}{{12}}{{\left( {\frac{5}{6}} \right)}^5}} \right)}
Question 15. Find the probability of throwing at most 2 sixes in 6 throws of a single die. {\displaystyle \left( {\frac{{35}}{{18}}{{\left( {\frac{5}{6}} \right)}^4}} \right)}
Question 16. 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? {\displaystyle \left( {\frac{{22 \times {9^3}}}{{{{10}^{11}}}}} \right)}
Question 17. In a box containing 100 bulbs, 10 are defective. Find the probability that out of a sample of 5 bulbs, none is defective. {\displaystyle \left( {{{\left( {\frac{9}{{10}}} \right)}^5}} \right)}
Question 18. The probability that a student is not a swimmer is {\displaystyle \frac{1}{5}} . Then find the probability that out of five students, four are swimmers. {\displaystyle \left( {^5{C_4}{{\left( {\frac{4}{5}} \right)}^4}\frac{1}{5}} \right)}

### Assignment – 5

Question 1. Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of the number of aces.
Question 2. Find the probability distribution of number of doublets in three throws of a pair of dice.
Question 3. Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values x, has the following form, where k is some unknown constant {\displaystyle {\rm{P(X)}} = \left\{ \begin{array}{l} 0.1 & {\rm{if}}\,x = 0\\ kx & {\rm{if}}\,x = 1 or 2\\ k(5-x) & {\rm{if}}\,x = 3 or 4\\ 0 & {\rm{otherwise}} \end{array} \right.}
Find the value of k. What is the probability that you study at least two hours ? Exactly two hours? At most two hours?
Question 4. Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X.
Question 5. Find the variance of the number obtained on a throw of an unbiased die. {\displaystyle \left( {\frac{{35}}{{12}}} \right)}
Question 6. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings. {\displaystyle (0.37)}
Question 7. Find the probability distribution of
a. number of heads in two tosses of a coin.
b. number of tails in the simultaneous tosses of three coins.
c. number of heads in four tosses of a coin
Question 8. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Question 9. A coin is biased 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.
Question 10. A random variable X has the following probability distribution:
X 0 1 2 3 4 5 6 7
P(X) 0 k 2 k 2 k 3 k {\displaystyle {k^2}} {\displaystyle 2{k^2}} {\displaystyle 7{k^2} + k}

Determine (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(0 < X < 3)
Question 11. The random variable X has a probability distribution P(X) of the following form, where k is some number : {\displaystyle {\rm{P(X)}} = \left\{ \begin{array}{l} k & {\rm{if}}\,x = 0\\ 2k & {\rm{if}}\,x = 1\\ 3k & {\rm{if}}\,x = 2\\ 0 & {\rm{otherwise}} \end{array} \right.}
Determine the value of k. Find P (X < 2), P (X ≤ 2), P(X ≥ 2).
Question 12. Find the mean number of heads in three tosses of a fair coin. {\displaystyle (1.5)}
Question 13. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X. {\displaystyle \left( {\frac{1}{3}} \right)}
Question 14. 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). {\displaystyle \left( {\frac{{14}}{3}} \right)}
Question 15. Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. {\displaystyle (5.833,\,\,2.415)}
Question 16. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of 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. {\displaystyle (17.53,\,\,4.78,\,\,2.19)}
Question 17. 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 Var (X). {\displaystyle (0.7,\,\,0.21)}
Question 18. Find the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face. {\displaystyle (2)}
Question 19. Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Find the value of E(X). {\displaystyle \left( {\frac{2}{{13}}} \right)}

### Assignment – 6

Question 1. A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time. {\displaystyle (0.488)}
Question 2. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red? {\displaystyle \left( {\frac{1}{2}} \right)}
Question 3. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II. {\displaystyle \left( {\frac{{35}}{{68}}} \right)}
Question 4. Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold? {\displaystyle \left( {\frac{2}{3}} \right)}
Question 5. 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. {\displaystyle \left( {\frac{2}{3}} \right)}
Question 6. 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 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 resides in hostel? {\displaystyle \left( {\frac{9}{{13}}} \right)}
Question 7. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ? {\displaystyle \left( {\frac{4}{9}} \right)}
Question 8. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? {\displaystyle \left( {\frac{1}{{52}}} \right)}
Question 9. 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 B 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? {\displaystyle \left( {\frac{1}{4}} \right)}
Question 10. 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, if the first group wins, the probability of introducing 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. {\displaystyle \left( {\frac{2}{9}} \right)}
Question 11. Suppose a girl throws a die. 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 die? {\displaystyle \left( {\frac{8}{{11}}} \right)}
Question 12. 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? {\displaystyle \left( {\frac{5}{{34}}} \right)}
Question 13. In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B? {\displaystyle \left( {\frac{{28}}{{69}}} \right)}
Question 14. A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively {\displaystyle \frac{3}{{10}},\,\,\frac{1}{5},\,\,\frac{1}{{10}}\,\& \,\frac{2}{5}} . The probabilities that he will be late are {\displaystyle \frac{1}{4},\,\,\frac{1}{3}\,\,\& \,\,\frac{1}{{12}}} if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he comes by train? {\displaystyle \left( {\frac{1}{2}} \right)}

### Assignment – 7

Question 1. A bag contains 10 white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black? {\displaystyle \left( {\frac{1}{4}} \right)}
Question 2. Find the probability of drawing a diamond card in each of the two consecutive draws from a well shuffled pack of cards, if the card drawn is not replaced after the first draw. {\displaystyle \left( {\frac{1}{{17}}} \right)}
Question 3. A bag contains 5 white, 7 red and 8 black balls. If four balls are drawn one by one without replacement, find the probability of getting all white balls. {\displaystyle \left( {\frac{1}{{969}}} \right)}
Question 4. A bag contains 19 tickets, numbered from 1 to 19. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both the tickets will show even numbers. {\displaystyle \left( {\frac{4}{{19}}} \right)}
Question 5. Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red? {\displaystyle \left( {\frac{7}{{12}}} \right)}
Question 6. Two card are drawn without replacement from a pack of 52 cards. Find the probability that
a. Both are kings
b. The first is king and second is an ace
c. The first is heart and second is red. {\displaystyle \left( {\frac{1}{{221}},\frac{4}{{663}},\frac{{25}}{{204}}} \right)}
Question 7. A bag contains 5 white, 7 red and 3 black balls . If three balls are drawn one by one without replacement, find the probability that none is red. {\displaystyle \left( {\frac{8}{{65}}} \right)}
Question 8. Three machines {\displaystyle {E_1},{E_2},{E_3}} in a certain factory produce 50%, 25% and 25% respectively, of the total daily output of the electric tubes. It is known that 4% of the tubes produced on each of the machines {\displaystyle {E_1}\& {E_2}} are defective and that of 5% of those produced on {\displaystyle {E_3}} are defective. If one tube is picked at random from a day’s production, calculate the probability that it is defective. (0.0425)
Question 9. Suppose 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 O. If a left handed person is selected at random, what is the probability that he/she will have blood group O? {\displaystyle \left( {\frac{9}{{44}}} \right)}
Question 10. By examining the chest X ray, the probability that 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? {\displaystyle \left( {\frac{{110}}{{221}}} \right)}
Question 11. A bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white. {\displaystyle \left( {\frac{{29}}{{63}}} \right)}
Question 12. The contents of three bags are as follows:
Bag I: 1 white, 2 black and 3 red balls
Bag II: 2 white , 1 black and 1 red ball
Bag III: 4 white, 5 black and 3 red balls
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red? {\displaystyle \left( {\frac{{118}}{{495}}} \right)}
Question 13. A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters are ON are visible. What is the probability that the letter has come from (i) LONDON (ii) CLIFTON? {\displaystyle \left( {\frac{{12}}{{17}},\,\,\frac{5}{{17}}} \right)}
Question 15. A drunkard man takes a step forward with probability 0.6 and takes a step backward with probability 0.4. He takes 9 steps in all. Find the probability that he is just one step away from the initial point. Do you think drinking habit can ruin one’s family life? {\displaystyle \left( {126\,\,{{(0.24)}^4}} \right)}
Question 16. For A, B and C the chances of being selected as the manager of a firm are in the ratio 4:1:2 respectively. The respective probabilities for them to introduce a radical change in marketing strategy are 0.3, 0.8 and 0.5. If the change does take place, find the probability that it is due to the appointment of B or C. {\displaystyle \left( {\frac{3}{5}} \right)}
Question 17. Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. Two balls are transferred at random 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 balls were both are black. {\displaystyle \left( {\frac{4}{{17}}} \right)}

### Assignment – 8

Question 1. Prove that if E and F are independent events, then so are the events E and F′.
Question 2. If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1– P(A′) P(B′)
Question 3. A card from a pack of 52 cards 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. {\displaystyle \left( {\frac{{11}}{{50}}} \right)}
Question 4. In answering a question on a multiple choice test, a student either knows the answer or guesses. Let {\displaystyle \frac{3}{4}} be the probability that he knows the answer and {\displaystyle \frac{1}{4}} be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability {\displaystyle \frac{1}{4}} . What is the probability that the student knows the answer given that he answered it correctly? {\displaystyle \left( {\frac{{12}}{{13}}} \right)}
Question 5. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. {\displaystyle \left( {\frac{3}{8}} \right)}
Question 6. Probability that A speaks truth is {\displaystyle \frac{4}{5}} . A coin is tossed. A reports that a head appears. Find the probability that actually there was head. {\displaystyle \left( {\frac{4}{5}} \right)}
Question 7. Suppose that the reliability of a HIV test is specified as follows:
Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the test are judged HIV–ive but 1% are diagnosed as showing HIV+ive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV+ive. What is the probability that the person actually has HIV? {\displaystyle \left( {\frac{{90}}{{1089}}} \right)}
Question 8. 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 disease). 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 ? {\displaystyle \left( {\frac{{198}}{{1197}}} \right)}
Question 9. 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 are equal number of males and females. {\displaystyle \left( {\frac{{20}}{{21}}} \right)}
Question 10. 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? {\displaystyle \left( {1 - \sum\limits_{r = 7}^{10} {^{10}{C_r}{{(0.9)}^r}{{(0.1)}^{10 - r}}} } \right)}
Question 11. In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is {\displaystyle \frac{5}{6}} . What is the probability that he will knock down fewer than 2 hurdles? {\displaystyle \left( {\frac{{{5^{10}}}}{{2 \times {6^9}}}} \right)}
Question 12. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die. {\displaystyle \left( {\frac{{625}}{{23328}}} \right)}
Question 13. An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes. {\displaystyle \left( {\frac{{31}}{9}{{\left( {\frac{2}{3}} \right)}^4}} \right)}
Question 14. How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%? {\displaystyle (n \ge 4)}
Question 15. The probability of a shooter hitting a target is {\displaystyle \frac{3}{4}} . How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99? {\displaystyle (4\,\,times)}
Question 16. 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 prescription 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 probability that the patient followed a course of meditation and yoga? {\displaystyle \left( {\frac{{14}}{{29}}} \right)}
Question 17. A and B throw a die alternatively till one of them gets a ‘6’ and wins the game. Find their respective probabilities of winning, if A starts first. {\displaystyle \left( {\frac{6}{{11}},\,\,\frac{5}{{11}}} \right)}
Question 18. If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup. {\displaystyle (0.95)}

Syllabus medium English Hinglish (Hindi + English) 12 Online learning Self-learning from videos Mathematics