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Mathematics · Class 12 Science

Ch 13Probability — concept-first.

The study of probability began as a way to measure uncertainty in random experiments. In earlier classes, you met the axiomatic approach to probability, developed by the Russian mathematician A.N.

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Concepts

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Key concepts

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Conditional Probability

Roll a die and ask "what is the chance of an even number?" — that is . But suppose someone tells you the result is greater than 3.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

13.1

Introduction

The study of probability began as a way to measure uncertainty in random experiments. In earlier classes, you met the axiomatic approach to probability, developed by the Russian mathematician A.N.

13.2

Conditional Probability

So far we have found probabilities of events using the entire sample space. But real life often gives us partial information: if we know one event has already occurred, does that change the chance of…

13.2.1

Properties of Conditional Probability

24 Q

Before we dive into the properties, recall what conditional probability means. When we write , we are asking: If we already know that event F has happened, what is the probability that event E also ha…

+Worked Examplesi7 questions
  1. Example 1If $P(A) = \frac{7}{13}$, $P(B) = \frac{9}{13}$ and $P(A \cap B) = \frac{4}{13}$, evaluate $P(A|B)$.Free
  2. Example 2A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?Free
  3. Example 3Ten 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 t…Free
  4. Example 4In 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…Preview
  5. Example 5A 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 th…Preview
  6. Example 6A 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 ap…Preview
  7. Example 7Consider 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 conditiona…Preview
+Exercise 13.1i17 questions
  1. Q1Given that E and F are events such that $P(E) = 0.6$, $P(F) = 0.3$ and $P(E \cap F) = 0.2$, find $P(E|F)$ and $P(F|E)$.Free
  2. Q2Compute $P(A|B)$, if $P(B) = 0.5$ and $P(A \cap B) = 0.32$.Free
  3. Q3If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (i) $P(A \cap B)$ (ii) $P(A|B)$ (iii) $P(A \cup B)$Free
  4. Q4Evaluate $P(A \cup B)$, if $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$.Preview
  5. Q5If $P(A) = \frac{6}{11}$, $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (i) $P(A \cap B)$ (ii) $P(A|B)$ (iii) $P(B|A)$ Determ…Preview
  6. Q6Determine $P(E|F)$. A coin is tossed three times, where (i) $E$ : head on third toss, $F$ : heads on first two tosses (ii) $E$ : at least tw…Preview
  7. Q7Determine $P(E|F)$. Two coins are tossed once, where (i) $E$ : tail appears on one coin, $F$ : one coin shows head (ii) $E$ : no tail appear…Preview
  8. Q8Determine $P(E|F)$. A die is thrown three times, $E$ : 4 appears on the third toss, $F$ : 6 and 5 appears respectively on first two tosses.Preview
  9. Q9Determine $P(E|F)$. Mother, father and son line up at random for a family picture. $E$ : son on one end, $F$ : father in middle.Preview
  10. Q10A 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…Preview
  11. Q11A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$ Find (i) $P(E|F)$ and $P(F|E)$ (ii) $P(E|G)$ and…Preview
  12. Q12Assume 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…Preview
  13. Q13An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple cho…Preview
  14. Q14Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice i…Preview
  15. Q15Consider 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. Fi…Preview
  16. Q16If $P(A) = \frac{1}{2}$, $P(B) = 0$, then $P(A|B)$ is (A) 0 (B) $\frac{1}{2}$ (C) not defined (D) 1Preview
  17. Q17If A and B are events such that $P(A|B) = P(B|A)$, then (A) $A \subset B$ but $A \neq B$ (B) $A = B$ (C) $A \cap B = \phi$ (D) $P(A) = P(B)$Preview
13.3

Multiplication Theorem on Probability

When two events and belong to the same sample space , the event that both occur is denoted , often written .

+Worked Examplesi2 questions
  1. Example 8An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probabilit…Free
  2. Example 9Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards…Preview
13.4

Independent Events

23 Q

Draw a single card from a well-shuffled deck of 52. Let be "the card is a spade" and be "the card is an ace." With 13 spades and 4 aces:

+Worked Examplesi5 questions
  1. Example 10A die is thrown. If $E$ is the event 'the number appearing is a multiple of 3' and $F$ be the event 'the number appearing is even' then find…Free
  2. Example 11An unbiased die is thrown twice. Let the event $A$ be 'odd number on the first throw' and $B$ the event 'odd number on the second throw'. Ch…Free
  3. Example 12Three coins are tossed simultaneously. Consider the event $E$ 'three heads or three tails', $F$ 'at least two heads' and $G$ 'at most two he…Preview
  4. Example 13Prove that if $E$ and $F$ are independent events, then so are the events $E$ and $F'$.Preview
  5. Example 14If $A$ and $B$ are two independent events, then the probability of occurrence of at least one of $A$ and $B$ is given by $1 - P(A')\,P(B')$.Preview
+Exercise 13.2i18 questions
  1. Q1If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$, find $P(A \cap B)$ if A and B are independent events.Free
  2. Q2Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.Free
  3. Q3A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the…Free
  4. Q4A fair coin and an unbiased die are tossed. Let $A$ be the event 'head appears on the coin' and $B$ be the event '3 on the die'. Check wheth…Preview
  5. Q5A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let $A$ be the event 'the number is even,' and $B$ be the event, 'the number is…Preview
  6. Q6Let E and F be events with $P(E) = \frac{3}{5}$, $P(F) = \frac{3}{10}$ and $P(E \cap F) = \frac{1}{5}$. Are E and F independent?Preview
  7. Q7Given that the events A and B are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = p$. Find $p$ if they are (i) mutua…Preview
  8. Q8Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find (i) $P(A \cap B)$ (ii) $P(A \cup B)$ (iii) $P(A|B)$ (iv) $P(B|A)$Preview
  9. Q9If A and B are two events such that $P(A) = \frac{1}{4}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{8}$, find P (not A and not B).Preview
  10. Q10Events A and B are such that $P(A) = \frac{1}{2}$, $P(B) = \frac{7}{12}$ and P(not A or not B) = $\frac{1}{4}$. State whether A and B are in…Preview
  11. Q11Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (i) $P(A \text{ and } B)$ (ii) $P(A \text{ and not } B)$ (ii…Preview
  12. Q12A die is tossed thrice. Find the probability of getting an odd number at least once.Preview
  13. Q13Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are…Preview
  14. Q14Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the…Preview
  15. Q15One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events $E$ and $F$ independent? (…Preview
  16. Q16In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student…Preview
  17. Q17Find the value of the following: The probability of obtaining an even prime number on each die, when a pair of dice is rolled is (A) 0 (B) $…Preview
  18. Q18Two events A and B will be independent, if (A) A and B are mutually exclusive (B) $P(A'B') = [1 - P(A)] [1 - P(B)]$ (C) $P(A) = P(B)$ (D) $P…Preview
13.5

Bayes' Theorem

To understand Bayes' Theorem we first see how a sample space can be broken into useful pieces. Consider a random experiment with sample space and events that are:

13.5.1

Partition of a Sample Space

Rolling a die gives sample space . Consider:

13.5.2

Theorem of Total Probability

21 Q

When we want to find the probability of an event , we often know how behaves under different conditions or scenarios.

+Worked Examplesi7 questions
  1. Example 15A person has undertaken a construction job. The probabilities are 0.65 that there will be a strike, 0.80 that the construction job will be c…Free
  2. Example 16Bag 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 ba…Free
  3. Example 17Given 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…Free
  4. Example 18Suppose 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 unde…Preview
  5. Example 19In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 an…Preview
  6. Example 20A 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…Preview
  7. Example 21A 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…Preview
+Exercise 13.3i14 questions
  1. Q1An 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 b…Free
  2. Q2A 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 i…Free
  3. Q3Of 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…Free
  4. Q4In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that h…Preview
  5. Q5A laboratory blood test is 99% effective in detecting a certain disease when it is in fact present. However, the test also yields a false po…Preview
  6. Q6There 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…Preview
  7. Q7An insurance company insured $2000$ scooter drivers, $4000$ car drivers and $6000$ truck drivers. The probability of an accidents are $0.01$…Preview
  8. Q8A 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…Preview
  9. Q9Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups…Preview
  10. Q10Suppose 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, sh…Preview
  11. Q11A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B…Preview
  12. Q12A 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 th…Preview
  13. Q13Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was h…Preview
  14. Q14If $A$ and $B$ are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct? (A) $P(A|B) = \frac{P(B)}{P…Preview

Miscellaneous

16 Q
+Miscellaneous Examplesi3 questions
  1. Example 22Coloured balls are distributed in four boxes as shown in the following table: | Box | Black | White | Red | Blue | | --- | --- | --- | --- |…Free
  2. Example 23$A$ and $B$ throw a die alternatively till one of them gets a '6' and wins the game. Find their respective probabilities of winning, if $A$…Preview
  3. Example 24If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past…Preview
+Miscellaneous Exercisei13 questions
  1. Q1A and B are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (i) A is a subset of B (ii) $A \cap B = \phi$Free
  2. Q2A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male.…Free
  3. Q3Suppose 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…Free
  4. Q4Suppose 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?Preview
  5. Q5If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?Preview
  6. Q6Suppose we have four boxes A,B,C and D containing coloured marbles as given below: $$ \begin{array}{|c|c|c|c|} \hline \text{Box} & \multicol…Preview
  7. Q7Assume 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 o…Preview
  8. Q8If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive?…Preview
  9. Q9An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed t…Preview
  10. Q10Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a…Preview
  11. Q11If A and B are two events such that $P(A) \neq 0$ and $P(B | A) = 1$, then (A) $A \subset B$ (B) $B \subset A$ (C) $B = \phi$ (D) $A = \phi$Preview
  12. Q12If $P(A|B) > P(A)$, then which of the following is correct : (A) $P(B|A) < P(B)$ (B) $P(A \cap B) < P(A) \cdot P(B)$ (C) $P(B|A) > P(B)$ (D)…Preview
  13. Q13If A and B are any two events such that $P(A) + P(B) – P(A \text{ and } B) = P(A)$, then (A) $P(B|A) = 1$ (B) $P(A|B) = 1$ (C) $P(B|A) = 0$…Preview

Summary

- Conditional Probability: , for . It redefines the sample space to .

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 70 questions70 questions
  1. Q1For a loaded die, the probabilities of outcomes are given as under: $P(1) = P(2) = 0.2$, $P(3) = P(5) = P(6) = 0.1$ and $P(4) = 0.3$. The di…Free
  2. Q2Refer to Exercise 1 above. If the die were fair, determine whether or not the events $A$ and $B$ are independent.Free
  3. Q3The probability that at least one of the two events $A$ and $B$ occurs is $0.6$. If $A$ and $B$ occur simultaneously with probability $0.3$,…Free
  4. Q4A bag contains $5$ red marbles and $3$ black marbles. Three marbles are drawn one by one without replacement. What is the probability that a…Preview
  5. Q5Two dice are thrown together and the total score is noted. The events $E$, $F$ and $G$ are 'a total of $4$', 'a total of $9$ or more', and '…Preview
  6. Q7$A$ and $B$ are two events such that $P(A) = \dfrac{1}{2}$, $P(B) = \dfrac{1}{3}$ and $P(A \cap B) = \dfrac{1}{4}$. Find: (i) $P(A \mid B)$…Preview
  7. Q8Three events $A$, $B$ and $C$ have probabilities $\dfrac{2}{5}$, $\dfrac{1}{3}$ and $\dfrac{1}{2}$, respectively. Given that $P(A \cap C) =…Preview
  8. Q9Let $E_1$ and $E_2$ be two independent events such that $P(E_1) = p_1$ and $P(E_2) = p_2$. Describe in words of the events whose probabiliti…Preview
  9. Q11Prove that (i) $P(A) = P(A \cap B) + P(A \cap B')$ (ii) $P(A \cup B) = P(A \cap B) + P(A \cap B') + P(A' \cap B)$Preview
  10. Q14Three dice are thrown at the same time. Find the probability of getting three two's, if it is known that the sum of the numbers on the dice…Preview
  11. Q16A bag contains $4$ white and $5$ black balls. Another bag contains $9$ white and $7$ black balls. A ball is transferred from the first bag t…Preview
  12. Q17Bag I contains $3$ black and $2$ white balls, Bag II contains $2$ black and $4$ white balls. A bag and a ball is selected at random. Determi…Preview
  13. Q18A box has $5$ blue and $4$ red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn…Preview
  14. Q19Four cards are successively drawn without replacement from a deck of $52$ playing cards. What is the probability that all the four cards are…Preview
  15. Q32Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is $2$-headed. If you take one out, toss…Preview
  16. Q33Suppose 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 t…Preview
  17. Q34Two natural numbers $r$, $s$ are drawn one at a time, without replacement from the set $S = \{1, 2, 3, \ldots, n\}$. Find $P[\,r \le p \mid…Preview
  18. Q38$A$ and $B$ throw a pair of dice alternately. $A$ wins the game if he gets a total of $6$ and $B$ wins if she gets a total of $7$. If $A$ st…Preview
  19. Q39Two dice are tossed. Find whether the following two events $A$ and $B$ are independent: $A = \{(x, y) : x + y = 11\}$, $B = \{(x, y) : x \ne…Preview
  20. Q40An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with $k$ additional balls of the…Preview
  21. Q41Three bags contain a number of red and white balls as follows: Bag 1: $3$ red balls, Bag 2: $2$ red balls and $1$ white ball, Bag 3: $3$ whi…Preview
  22. Q42Refer to Question 41 above. If a white ball is selected, what is the probability that it came from (i) Bag 2 (ii) Bag 3.Preview
  23. Q43A shopkeeper sells three types of flower seeds $A_1$, $A_2$ and $A_3$. They are sold as a mixture where the proportions are $4:4:2$ respecti…Preview
  24. Q44A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letters TA are visible. What i…Preview
  25. Q45There are two bags, one of which contains $3$ black and $4$ white balls while the other contains $4$ black and $3$ white balls. A die is thr…Preview
  26. Q46There are three urns containing $2$ white and $3$ black balls, $3$ white and $2$ black balls, and $4$ white and $1$ black balls, respectivel…Preview
  27. Q47By examining the chest X-ray, the probability that TB is detected when a person is actually suffering is $0.99$. The probability of a health…Preview
  28. Q48An item is manufactured by three machines $A$, $B$ and $C$. Out of the total number of items manufactured during a specified period, $50\%$…Preview
  29. Q52A bag contains $(2n + 1)$ coins. It is known that $n$ of these coins have a head on both sides whereas the rest of the coins are fair. A coi…Preview
  30. Q56If $P(A) = \dfrac{4}{5}$ and $P(A \cap B) = \dfrac{7}{10}$, then $P(B \mid A)$ is equal to (A) $\dfrac{1}{10}$ (B) $\dfrac{1}{8}$ (C) $\dfra…Preview
  31. Q57If $P(A \cap B) = \dfrac{7}{10}$ and $P(B) = \dfrac{17}{20}$, then $P(A \mid B)$ equals (A) $\dfrac{14}{17}$ (B) $\dfrac{17}{20}$ (C) $\dfra…Preview
  32. Q58If $P(A) = \dfrac{3}{10}$, $P(B) = \dfrac{2}{5}$ and $P(A \cup B) = \dfrac{3}{5}$, then $P(B \mid A) + P(A \mid B)$ equals (A) $\dfrac{1}{4}…Preview
  33. Q59If $P(A) = \dfrac{2}{5}$, $P(B) = \dfrac{3}{10}$ and $P(A \cap B) = \dfrac{1}{5}$, then $P(A' \mid B') \cdot P(B' \mid A')$ is equal to (A)…Preview
  34. Q60If $A$ and $B$ are two events such that $P(A) = \dfrac{1}{2}$, $P(B) = \dfrac{1}{3}$, $P(A \mid B) = \dfrac{1}{4}$, then $P(A' \cap B')$ equ…Preview
  35. Q61If $P(A) = 0.4$, $P(B) = 0.8$ and $P(B \mid A) = 0.6$, then $P(A \cup B)$ is equal to (A) $0.24$ (B) $0.3$ (C) $0.48$ (D) $0.96$Preview
  36. Q62If $A$ and $B$ are two events and $A \neq \varphi$, $B \neq \varphi$, then (A) $P(A \mid B) = P(A) \cdot P(B)$ (B) $P(A \mid B) = \dfrac{P(A…Preview
  37. Q63$A$ and $B$ are events such that $P(A) = 0.4$, $P(B) = 0.3$ and $P(A \cup B) = 0.5$. Then $P(B' \cap A)$ equals (A) $\dfrac{2}{3}$ (B) $\dfr…Preview
  38. Q64You are given that $A$ and $B$ are two events such that $P(B) = \dfrac{3}{5}$, $P(A \mid B) = \dfrac{1}{2}$ and $P(A \cup B) = \dfrac{4}{5}$…Preview
  39. Q65In Exercise 64 above, $P(B \mid A')$ is equal to (A) $\dfrac{1}{5}$ (B) $\dfrac{3}{10}$ (C) $\dfrac{1}{2}$ (D) $\dfrac{3}{5}$Preview
  40. Q66If $P(B) = \dfrac{3}{5}$, $P(A \mid B) = \dfrac{1}{2}$ and $P(A \cup B) = \dfrac{4}{5}$, then $P((A \cup B)') + P(A' \cup B)$ equals (A) $\d…Preview
  41. Q67Let $P(A) = \dfrac{7}{13}$, $P(B) = \dfrac{9}{13}$ and $P(A \cap B) = \dfrac{4}{13}$. Then $P(A' \mid B)$ is equal to (A) $\dfrac{6}{13}$ (B…Preview
  42. Q68If $A$ and $B$ are such events that $P(A) > 0$ and $P(B) \neq 1$, then $P(A' \mid B')$ equals (A) $1 - P(A \mid B)$ (B) $1 - P(A' \mid B)$ (…Preview
  43. Q69If $A$ and $B$ are two independent events with $P(A) = \dfrac{3}{5}$ and $P(B) = \dfrac{4}{9}$, then $P(A' \cap B')$ equals (A) $\dfrac{4}{1…Preview
  44. Q70If two events are independent, then (A) they must be mutually exclusive (B) the sum of their probabilities must be equal to $1$ (C) (A) and…Preview
  45. Q71Let $A$ and $B$ be two events such that $P(A) = \dfrac{3}{8}$, $P(B) = \dfrac{5}{8}$ and $P(A \cup B) = \dfrac{3}{4}$. Then $P(A \mid B) \cd…Preview
  46. Q72If the events $A$ and $B$ are independent, then $P(A \cap B)$ is equal to (A) $P(A) + P(B)$ (B) $P(A) - P(B)$ (C) $P(A) \cdot P(B)$ (D) $\df…Preview
  47. Q73Two events $E$ and $F$ are independent. If $P(E) = 0.3$, $P(E \cup F) = 0.5$, then $P(E \mid F) - P(F \mid E)$ equals (A) $\dfrac{2}{7}$ (B)…Preview
  48. Q74A bag contains $5$ red and $3$ blue balls. If $3$ balls are drawn at random without replacement, the probability of getting exactly one red…Preview
  49. Q75Refer to Question 74 above. The probability that exactly two of the three balls were red, the first ball being red, is (A) $\dfrac{1}{3}$ (B…Preview
  50. Q76Three persons, $A$, $B$ and $C$, fire at a target in turn, starting with $A$. Their probabilities of hitting the target are $0.4$, $0.3$ and…Preview
  51. Q77Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probabili…Preview
  52. Q78A die is thrown and a card is selected at random from a deck of $52$ playing cards. The probability of getting an even number on the die and…Preview
  53. Q79A box contains $3$ orange balls, $3$ green balls and $2$ blue balls. Three balls are drawn at random from the box without replacement. The p…Preview
  54. Q80A flashlight has $8$ batteries out of which $3$ are dead. If two batteries are selected without replacement and tested, the probability that…Preview
  55. Q82Two dice are thrown. If it is known that the sum of numbers on the dice was less than $6$, the probability of getting a sum $3$ is (A) $\dfr…Preview
  56. Q84Two cards are drawn from a well shuffled deck of $52$ playing cards with replacement. The probability that both cards are queens is (A) $\df…Preview
  57. Q91In a college, $30\%$ students fail in physics, $25\%$ fail in mathematics and $10\%$ fail in both. One student is chosen at random. The prob…Preview
  58. Q92$A$ and $B$ are two students. Their chances of solving a problem correctly are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. If the proba…Preview
  59. Q94Fill in the blank: If $A$ and $B$ are two events such that $P(A \mid B) = p$, $P(A) = p$, $P(B) = \dfrac{1}{3}$ and $P(A \cup B) = \dfrac{5}…Preview
  60. Q95Fill in the blank: If $A$ and $B$ are such that $P(A' \cup B') = \dfrac{2}{3}$ and $P(A \cup B) = \dfrac{5}{9}$, then $P(A') + P(B') = $ ___…Preview
  61. Q98Fill in the blank: Let $A$ and $B$ be two events. If $P(A \mid B) = P(A)$, then $A$ is __________ of $B$.Preview
  62. Q99State whether the following statement is True or False: Let $P(A) > 0$ and $P(B) > 0$. Then $A$ and $B$ can be both mutually exclusive and i…Preview
  63. Q100State whether the following statement is True or False: If $A$ and $B$ are independent events, then $A'$ and $B'$ are also independent.Preview
  64. Q101State whether the following statement is True or False: If $A$ and $B$ are mutually exclusive events, then they will be independent also.Preview
  65. Q102State whether the following statement is True or False: Two independent events are always mutually exclusive.Preview
  66. Q103State whether the following statement is True or False: If $A$ and $B$ are two independent events then $P(A \text{ and } B) = P(A) \cdot P(B…Preview
  67. Q105State whether the following statement is True or False: If $A$ and $B'$ are independent events, then $P(A' \cup B) = 1 - P(A)\,P(B')$.Preview
  68. Q106State whether the following statement is True or False: If $A$ and $B$ are independent, then $P(\text{exactly one of } A, B \text{ occurs})…Preview
  69. Q107State whether the following statement is True or False: If $A$ and $B$ are two events such that $P(A) > 0$ and $P(A) + P(B) > 1$, then $P(B…Preview
  70. Q108State whether the following statement is True or False: If $A$, $B$ and $C$ are three independent events such that $P(A) = P(B) = P(C) = p$,…Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 2 questions2 questions
  1. Q1A person observed the first 4 digits of your 6-digit PIN. What is the probability that the person can guess your PIN? (A) 1 81 (B) 1 100 (C)…Preview
  2. Q2For any two events $A$ and $B$, if $P(A) = \frac{1}{2}$, $P(B) = \frac{2}{3}$ and $P(A \cap B) = \frac{1}{4}$, then $P(\bar{A}/\bar{B})$ equ…Preview