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Mathematics · Class 11 Science

Ch 14Probability — Class 11 Mathematics, concept-first.

The sample space of a random experiment is the set of all possible outcomes. But when we actually perform the experiment, we are rarely interested in every outcome individually. Instead, we care about whether a particular happening occurs — for example, "getting exactly one head" when tossing a coin twice.

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Think of your two favourite groups of friends — the ones who play cricket and the ones who play football. Some friends are in both groups, some in only one, and some in neither.

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14.1

Event

The sample space of a random experiment is the set of all possible outcomes. But when we actually perform the experiment, we are rarely interested in every outcome individually.

14.1.1

Occurrence of an Event

When we talk about an event in probability, we are not just naming a possibility — we are describing a specific collection of outcomes from the sample space.

14.1.2

Types of Events

Events are subsets of the sample space, and the number of elements they contain determines their classification.

14.1.3

Algebra of Events

In the chapter on sets, you learned how to combine sets using operations like union, intersection, difference, and complement.

+Worked Examplesi1 question
  1. Example 1Consider the experiment of rolling a die. Let A be the event 'getting a prime number', B be the event 'getting an odd number'. Write the set…Preview
14.1.4

Mutually Exclusive Events

When you roll a fair die, the sample space is . Now consider two events:

14.1.5

Exhaustive Events

9 Q

When we throw a die, the sample space is . Consider three events:

+Worked Examplesi2 questions
  1. Example 2Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this…Free
  2. Example 3A coin is tossed three times, consider the following events. A: 'No head appears', B: 'Exactly one head appears' and C: 'Atleast two heads a…Preview
+Exercise 14.1i7 questions
  1. Q1A die is rolled. Let E be the event 'die shows 4' and F be the event 'die shows even number'. Are E and F mutually exclusive?Free
  2. Q2A die is thrown. Describe the following events: (i) A: a number less than 7 (ii) B: a number greater than 7 (iii) C: a multiple of 3 (iv) D:…Free
  3. Q3An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater t…Free
  4. Q4Three coins are tossed once. Let A denote the event 'three heads show', B denote the event 'two heads and one tail show', C denote the event…Preview
  5. Q5Three coins are tossed. Describe (i) Two events which are mutually exclusive. (ii) Three events which are mutually exclusive and exhaustive.…Preview
  6. Q6Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first…Preview
  7. Q7Refer to question 6 above, state true or false: (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually e…Preview
14.2

Axiomatic Approach to Probability

In earlier sections, you learned about random experiments, sample spaces, and events. But how do we actually assign a number — a probability — to an event? The classical approach you may have seen bef…

+Worked Examplesi1 question
  1. Example 4Let a sample space be $S = \{\omega_1, \omega_2, \ldots, \omega_6\}$. Which of the following assignments of probabilities to each outcome ar…Preview
14.2.1

Probability of an Event

When we talk about the probability of an event, we are really asking: "What is the chance that the outcome of an experiment falls into a particular collection of outcomes?" That collection is the even…

14.2.2

Probabilities of Equally Likely Outcomes

When you toss a fair coin, you know that heads and tails are equally likely. But what does "equally likely" mean in a precise, mathematical sense? It means that every simple event in the sample space…

14.2.3

Probability of the Event ‘A or B’

When two events A and B are considered together, the event ‘A or B’ means that at least one of them occurs. In set notation, this is the union .

14.2.4

Probability of Event ‘not A’

25 Q

When we talk about the probability of an event, we often also need the probability that the event does not happen. This is the probability of the complementary event, written as “not A” or .

+Worked Examplesi4 questions
  1. Example 5One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be…Free
  2. Example 6A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random f…Free
  3. Example 7Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will…Preview
  4. Example 8A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man? (b) one…Preview
+Exercise 14.2i21 questions
  1. Q1Which of the following cannot be valid assignment of probabilities for outcomes of sample space $S = \{\omega_1, \omega_2, \omega_3, \omega_…Free
  2. Q2A coin is tossed twice, what is the probability that atleast one tail occurs?Free
  3. Q3A die is thrown, find the probability of following events: (i) A prime number will appear, (ii) A number greater than or equal to 3 will app…Free
  4. Q4A card is selected from a pack of 52 cards. (a) How many points are there in the sample space? (b) Calculate the probability that the card i…Preview
  5. Q5A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of numbers that t…Preview
  6. Q6There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it…Preview
  7. Q7A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up. From the sample space ca…Preview
  8. Q8Three coins are tossed once. Find the probability of getting (i) 3 heads (ii) 2 heads (iii) atleast 2 heads (iv) atmost 2 heads (v) no head…Preview
  9. Q9If $\frac{2}{11}$ is the probability of an event A, what is the probability of the event 'not A'.Preview
  10. Q10A letter is chosen at random from the word 'ASSASSINATION'. Find the probability that letter is (i) a vowel (ii) a consonant.Preview
  11. Q11In a lottery, a person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers alre…Preview
  12. Q12Check whether the following probabilities P(A) and P(B) are consistently defined (i) $P(A) = 0.5$, $P(B) = 0.7$, $P(A \cap B) = 0.6$ (ii) $P…Preview
  13. Q13Fill in the blanks in following table: | | $P(A)$ | $P(B)$ | $P(A \cap B)$ | $P(A \cup B)$ | |---|---|---|---|---| | (i) | $\frac{1}{3}$ | $…Preview
  14. Q14Given $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$. Find P(A or B), if A and B are mutually exclusive events.Preview
  15. Q15If E and F are events such that $P(E) = \frac{1}{4}$, $P(F) = \frac{1}{2}$ and P(E and F) $= \frac{1}{8}$, find (i) P(E or F), (ii) P(not E…Preview
  16. Q16Events E and F are such that P(not E or not F) = 0.25. State whether E and F are mutually exclusive.Preview
  17. Q17A and B are events such that $P(A) = 0.42$, $P(B) = 0.48$ and P(A and B) = 0.16. Determine (i) P(not A), (ii) P(not B) and (iii) P(A or B).Preview
  18. Q18In Class XI of a school 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If…Preview
  19. Q19In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examinat…Preview
  20. Q20The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.…Preview
  21. Q21In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at ran…Preview

Miscellaneous Examples

+Miscellaneous Examplesi4 questions
  1. Example 9On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B? (ii) A…Free
  2. Example 10Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards, it contains (i) all Kings (ii) 3 Kings (iii…Free
  3. Example 11If A, B, C are three events associated with a random experiment, prove that $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \ca…Preview
  4. Example 12In a relay race there are five teams A, B, C, D and E. (a) What is the probability that A, B and C finish first, second and third, respectiv…Preview

Miscellaneous Exercise on Chapter 14

+Miscellaneous Exercisei10 questions
  1. Q1A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that (i) all…Free
  2. Q24 cards are drawn from a well-shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and one spade?Free
  3. Q3A die has two faces each with number '1', three faces each with number '2' and one face with number '3'. If die is rolled once, determine (i…Free
  4. Q4In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy (a)…Preview
  5. Q5Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that (…Preview
  6. Q6Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at ran…Preview
  7. Q7A and B are two events such that $P(A) = 0.54$, $P(B) = 0.69$ and $P(A \cap B) = 0.35$. Find (i) $P(A \cup B)$ (ii) $P(A' \cap B')$ (iii) $P…Preview
  8. Q8From the employees of a company, 5 persons are selected to represent them in the managing committee of the company. Particulars of five pers…Preview
  9. Q9If 4-digit numbers greater than 5,000 are randomly formed from the digits 0, 1, 3, 5, and 7, what is the probability of forming a number div…Preview
  10. Q10The number lock of a suitcase has 4 wheels, each labelled with ten digits i.e., from 0 to 9. The lock opens with a sequence of four digits w…Preview

Summary

- Random experiment sample space: A random experiment has multiple possible outcomes; the set of all outcomes is the sample space . An event is any subset of .

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 43 questions43 questions
  1. Q1If the letters of the word ALGORITHM are arranged at random in a row, what is the probability the letters GOR must remain together as a unit…Free
  2. Q2Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of empl…Free
  3. Q3Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.Free
  4. Q4An experiment consists of rolling a die until a 2 appears. (i) How many elements of the sample space correspond to the event that the 2 appe…Preview
  5. Q5A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find $P(G)$, where $G$ is the event that…Preview
  6. Q6In a large metropolitan area, the probabilities are $.87, .36, .30$ that a family (randomly chosen for a sample survey) owns a colour televi…Preview
  7. Q7If $A$ and $B$ are mutually exclusive events, $P(A) = 0.35$ and $P(B) = 0.45$, find (a) $P(A')$ (b) $P(B')$ (c) $P(A \cup B)$ (d) $P(A \cap…Preview
  8. Q8A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated a…Preview
  9. Q9Four candidates A, B, C, D have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and…Preview
  10. Q10One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary…Preview
  11. Q11The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, $P(A \…Preview
  12. Q12One urn contains two black balls (labelled $B_1$ and $B_2$) and one white ball. A second urn contains one black ball and two white balls (la…Preview
  13. Q13A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that (a) All the three balls are white. (b) Al…Preview
  14. Q14If the letters of the word ASSASSINATION are arranged at random, find the probability that (a) Four S's come consecutively in the word. (b)…Preview
  15. Q15A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.Preview
  16. Q16A sample space consists of 9 elementary outcomes $e_1, e_2, \ldots, e_9$ whose probabilities are $P(e_1) = P(e_2) = .08$, $P(e_3) = P(e_4) =…Preview
  17. Q17Determine the probability $p$, for each of the following events. (a) An odd number appears in a single toss of a fair die. (b) At least one…Preview
  18. Q18In a non-leap year, the probability of having 53 tuesdays or 53 wednesdays is (A) $\frac{1}{7}$ (B) $\frac{2}{7}$ (C) $\frac{3}{7}$ (D) none…Preview
  19. Q19Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive (A) $\frac{186}{190}$ (B) $\frac{187}{190}$ (C) $\…Preview
  20. Q20While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colou…Preview
  21. Q21Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is (A) $\frac{1}{3}$ (B) $\frac{…Preview
  22. Q22Without repetition of the numbers, four digit numbers are formed with the numbers $0, 2, 3, 5$. The probability of such a number divisible b…Preview
  23. Q23If $A$ and $B$ are mutually exclusive events, then (A) $P(A) \leq P(\bar{B})$ (B) $P(A) \geq P(\bar{B})$ (C) $P(A) < P(\bar{B})$ (D) none of…Preview
  24. Q24If $P(A \cup B) = P(A \cap B)$ for any two events $A$ and $B$, then (A) $P(A) = P(B)$ (B) $P(A) > P(B)$ (C) $P(A) < P(B)$ (D) none of thesePreview
  25. Q256 boys and 6 girls sit in a row at random. The probability that all the girls sit together is (A) $\frac{1}{432}$ (B) $\frac{12}{431}$ (C) $…Preview
  26. Q26A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel is (A) $\frac{1}{3}$ (B) $\frac{4}{11}…Preview
  27. Q27If the probabilities for A to fail in an examination is $0.2$ and that for B is $0.3$, then the probability that either A or B fails is (A)…Preview
  28. Q28The probability that at least one of the events A and B occurs is $0.6$. If A and B occur simultaneously with probability $0.2$, then $P(\ba…Preview
  29. Q29If $M$ and $N$ are any two events, the probability that at least one of them occurs is (A) $P(M) + P(N) - 2P(M \cap N)$ (B) $P(M) + P(N) - P…Preview
  30. Q30The probability that the home team will win an upcoming football game is $0.77$, the probability that it will tie the game is $0.08$, and th…Preview
  31. Q31If $e_1, e_2, e_3, e_4$ are the four elementary outcomes in a sample space and $P(e_1) = .1$, $P(e_2) = .5$, $P(e_3) = .1$, then the probabi…Preview
  32. Q32Let $S = \{1, 2, 3, 4, 5, 6\}$ and $E = \{1, 3, 5\}$, then $\bar{E}$ is _________.Preview
  33. Q33If A and B are two events associated with a random experiment such that $P(A) = 0.3$, $P(B) = 0.2$ and $P(A \cap B) = 0.1$, then the value o…Preview
  34. Q34The probability of happening of an event A is $0.5$ and that of B is $0.3$. If A and B are mutually exclusive events, then the probability o…Preview
  35. Q35The probability that a person visiting a zoo will see the giraffee is $0.72$, the probability that he will see the bears is $0.84$ and the p…Preview
  36. Q36The probability that a student will pass his examination is $0.73$, the probability of the student getting a compartment is $0.13$, and the…Preview
  37. Q37The probabilities that a typist will make $0, 1, 2, 3, 4, 5$ or more mistakes in typing a report are, respectively, $0.12, 0.25, 0.36, 0.14,…Preview
  38. Q38If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is $.5$ and the probability th…Preview
  39. Q39The probability of intersection of two events A and B is always less than or equal to those favourable to the event A.Preview
  40. Q40The probability of an occurrence of event A is $.7$ and that of the occurrence of event B is $.3$ and the probability of occurrence of both…Preview
  41. Q41The sum of probabilities of two students getting distinction in their final examinations is $1.2$.Preview
  42. Q42Match the proposed probability under Column $C_1$ with the appropriate written description under column $C_2$: | $C_1$ (Probability) | $C_2$…Preview
  43. Q43Match the following: | Column I | Column II | | --- | --- | | (a) If $E_1$ and $E_2$ are the two mutually exclusive events | (i) $E_1 \cap E…Preview