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Mathematics · Class 11 Science

Ch 9Probability — Class 11 Mathematics, concept-first.

A random experiment is a trial that has more than one possible result: we know in advance every result that could occur, but we cannot say which one will actually happen. Rolling a die, tossing a coin, and drawing a card are all random experiments.

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Axiomatic and Classical Probability of an Event

For a random experiment whose outcomes are all equally likely (no outcome is preferred over any other), the classical definition assigns to an event the probability , the fraction of all outcomes that are favourable to .…

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Chapter contents

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9.1.1

Basic Terminologies

A random experiment is a trial that has more than one possible result: we know in advance every result that could occur, but we cannot say which one will actually happen.

9.1.2

Concept of Probability

Even though every possible outcome of a random experiment is known beforehand, which one will actually occur remains uncertain - that uncertainty is precisely what probability measures.

9.1.3

Equally Likely Outcomes

The outcomes of a random experiment are called equally likely if there is no reason to prefer any one outcome over any other - each is just as probable as the rest.

9.1.4

Probability of an Event

For a random experiment whose outcomes are equally likely, the probability of an event is defined as where is the number of outcomes favourable to (the outcomes belonging to the subset ) and is the to…

9.1.5

Elementary Properties of Probability

Building on the classical formula , several properties follow immediately and are used constantly in later sections:

9.2.1

Addition Theorem for Two Events

The addition theorem states that for any two events and of a sample space , This was already listed as property 7 in the previous section, but because it is used so heavily it deserves its own proof,…

9.3.1

Conditional Probability - Definition

So far every probability has been computed against the full sample space . Often, though, we already know something has happened, and we want the probability of a further event given that information.…

9.3.2

Formula for Conditional Probability

The two worked examples in the previous section suggest a general shortcut: instead of listing the reduced sample space every time, conditional probability can be written directly in terms of ordinary…

9.3.3

Multiplication Theorem

Rearranging the conditional probability formula for gives the multiplication theorem: for any two events and on a sample space , and symmetrically, from , also .

9.3.4

Independent Events

Two events and are called independent if the occurrence of either one does not change the probability of the other - knowing one has happened gives no information about the other.

9.4

Bayes' Theorem

Bayes' theorem (also called Bayes' law or Bayes' rule) is a direct extension of conditional probability that lets us go 'in reverse' - from knowing an outcome occurred, back to the probability that a…

9.5

Odds

Odds compare the two complementary probabilities of an event directly, as a ratio of favourable to unfavourable outcomes rather than as a fraction of the total.

More questions

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  1. Q65There are 5 girls and 2 boys, then the probability that no two boys are sitting together for a photograph is A) $\frac{1}{21}$ B) $\frac{4}{…Free
  2. Q66In a jar there are 5 black marbles and 3 green marbles. Two marbles are picked randomly one after the other without replacement. What is the…Free
  3. Q67Two dice are thrown simultaneously. Then the probability of getting two numbers whose product is even is A) $\frac{3}{4}$ B) $\frac{1}{4}$ C…Free
  4. Q68In a set of 30 shirts, 17 are white and rest are black. 4 white and 5 black shirts are tagged as 'PARTY WEAR'. If a shirt is chosen at rando…Preview
  5. Q69There are 2 shelves. One shelf has 5 Physics and 3 Biology books and the other has 4 Physics and 2 Biology books. The probability of drawing…Preview
  6. Q70Two friends $A$ and $B$ apply for a job in the same company. The chances of $A$ getting selected is 2/5 and that of $B$ is 4/7. The probabil…Preview
  7. Q71The probability that a student knows the correct answer to a multiple choice question is $\frac{2}{3}$. If the student does not know the ans…Preview
  8. Q72Bag 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…Preview
  9. Q73A fair die is tossed twice. What are the odds in favour of getting 4, 5 or 6 on the first toss and 1, 2, 3 or 4 on the second die? A) 1 : 3…Preview
  10. Q74The odds against an event are 5:3 and the odds in favour of another independent event are 7:5. The probability that at least one of the two…Preview
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  1. Q75The letters of the word 'EQUATION' are arranged in a row. Find the probability that a) All the vowels are together b) Arrangement starts wit…Free
  2. Q76There are 6 positive and 8 negative numbers. Four numbers are chosen at random, without replacement, and multiplied. Find the probability th…Free
  3. Q77Ten 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. Q78If $P(A\cap B) = \frac{1}{2}$, $P(B\cap C) = \frac{1}{3}$, $P(C\cap A) = \frac{1}{6}$ then find $P(A)$, $P(B)$ and $P(C)$, If $A,B,C$ are in…Preview
  5. Q79If the letters of the word 'REGULATIONS' be arranged at random, what is the probability that there will be exactly 4 letters between $R$ and…Preview
  6. Q80In how many ways can the letters of the word ARRANGEMENTS be arranged? a) Find the chance that an arrangement chosen at random begins with t…Preview
  7. Q81A letter is taken at random from the letters of the word 'ASSISTANT' and another letter is taken at random from the letters of the word 'STA…Preview
  8. Q82A die is loaded in such a way that the probability of the face with $j$ dots turning up is proportional to $j$ for $j = 1, 2, ..., 6$. What…Preview
  9. Q83An urn contains 5 red balls and 2 green balls. A ball is drawn. If it's green a red ball is added to the urn and if it's red a green ball is…Preview
  10. Q84The odds against $A$ solving a certain problem are 4 to 3 and the odds in favor of solving the same problem are 7 to 5, find the probability…Preview
  11. Q85If $P(A) = P\left(\frac{A}{B}\right) = \frac{1}{5}$, $P\left(\frac{B}{A}\right) = \frac{1}{3}$ then find (i) $P\left(\frac{A}{B'}\right)$ (i…Preview
  12. Q86Let $A$ and $B$ be independent events with $P(A) = \frac{1}{4}$, and $P(A\cup B) = 2P(B) - P(A)$. Find a) $P(B)$; b) $P(A/B)$; and c) $P(B'/…Preview
  13. Q87Find the probability that a year selected will have 53 Wednesdays.Preview
  14. Q88The chances of $P$, $Q$ and $R$, getting selected as principal of a college are $\frac{2}{5}$, $\frac{2}{5}$, $\frac{1}{5}$ respectively. Th…Preview
  15. Q89Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and…Preview
  16. Q90For three events $A$, $B$ and $C$, we know that $A$ and $C$ are independent, $B$ and $C$ are independent, $A$ and $B$ are disjoint, $P(A\cup…Preview
  17. Q91The ratio of Boys to Girls in a college is 3:2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer i…Preview
  18. Q92$A$ and $B$ throw a die alternately till one of them gets a 3 and wins the game. Find the respective probabilities of winning. (Assuming $A$…Preview
  19. Q93Consider independent trials consisting of rolling a pair of fair dice, over and over. What is the probability that a sum of 5 appears before…Preview
  20. Q94A machine produces parts that are either good (90%), slightly defective (2%), or obviously defective (8%). Produced parts get passed through…Preview
  21. Q95Given 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…Preview
  22. Q96In a factory which manufactures bulbs, machines $A$, $B$ and $C$ manufacture respectively 25%, 35% and 40% of the bulbs. Of their outputs, 5…Preview
  23. Q97A family has two children. One of them is chosen at random and found that the child is a girl. Find the probability that a) both the childre…Preview
+Show 17 questions17 questions
  1. Q1There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one after the other with replacem…Free
  2. Q2A coin and a die are tossed simultaneously. Enumerate the sample space and the following events. a) $A$: Getting a Tail and an Odd number. b…Free
  3. Q3Find $n(S)$ for each of the following random experiments. a) From an urn containing 5 gold and 3 silver coins, 3 coins are drawn at random.…Free
  4. Q4Two fair dice are thrown. State the sample space and write the favorable outcomes for the following events. a) $A$: Sum of numbers on two di…Preview
  5. Q5A bag contains four cards marked as 5, 6, 7 and 8. Find the sample space if two cards are drawn at random a) with replacement b) without rep…Preview
  6. Q6A fair die is thrown two times. Find the probability that a) sum of the numbers on them is 5 b) sum of the numbers on them is at least 8 c)…Preview
  7. Q7Two cards are drawn from a pack of 52 cards. Find the probability that a) one is a face card and the other is an ace card b) one is club and…Preview
  8. Q8Three cards are drawn from a pack of 52 cards. Find the chance that a) two are queen cards and one is an ace card b) at least one is a diamo…Preview
  9. Q9From a bag containing 10 red, 4 blue and 6 black balls, a ball is drawn at random. Find the probability of drawing a) a red ball b) a blue o…Preview
  10. Q10A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that, a) Number on the ticket is…Preview
  11. Q11What is the chance that a leap year, selected at random, will contain 53 Sundays?Preview
  12. Q12Find the probability of getting both red balls, when from a bag containing 5 red and 4 black balls, two balls are drawn, i) with replacement…Preview
  13. Q13A room has three sockets for lamps. From a collection 10 bulbs of which 6 are defective. At night a person selects 3 bulbs, at random and pu…Preview
  14. Q14Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement a) vowels are always together b) vowels are…Preview
  15. Q154 letters are to be posted in 4 post boxes. If any number of letters can be posted in any of the 4 post boxes, what is the probability that…Preview
  16. Q1615 professors have been invited for a round table conference by Vice chancellor of a university. What is the probability that two particular…Preview
  17. Q17A bag contains 7 black and 4 red balls. If 3 balls are drawn at random find the probability that (i) all are black (ii) one is black and two…Preview
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  1. Q18First 6 faced die which is numbered 1 through 6 is thrown then a 5 faced die which is numbered 1 through 5 is thrown. What is the probabilit…Free
  2. Q19A card is drawn from a pack of 52 cards. What is the probability that, i) card is either red or black? ii) card is either black or a face ca…Free
  3. Q20A girl is preparing for National Level Entrance exam and State Level Entrance exam for professional courses. The chances of her cracking Nat…Free
  4. Q21A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that, a) number on the ticket is a perf…Preview
  5. Q22The probability that a student will pass in French is 0.64, will pass in Sociology is 0.45 and will pass in both is 0.40. What is the probab…Preview
  6. Q23Two fair dice are thrown. Find the probability that number on the upper face of the first die is 3 or sum of the numbers on their upper face…Preview
  7. Q24For two events $A$ and $B$ of a sample space $S$, if $P(A) = \frac{3}{8}$, $P(B) = \frac{1}{2}$ and $P(A\cup B) = \frac{5}{8}$. Find the val…Preview
  8. Q25For two events $A$ and $B$ of a sample space $S$, if $P(A\cup B) = \frac{5}{6}$, $P(A\cap B) = \frac{1}{3}$ and $P(B') = \frac{1}{3}$, then…Preview
  9. Q26A bag contains 5 red, 4 blue and an unknown number $m$ of green balls. If the probability of getting both the balls green, when two balls ar…Preview
  10. Q27From a group of 4 men, 4 women and 3 children, 4 persons are selected at random. Find the probability that, i) no child is selected ii) exac…Preview
  11. Q28A number is drawn at random from the numbers 1 to 50. Find the probability that it is divisible by 2 or 3 or 10.Preview
+Show 18 questions18 questions
  1. Q29A bag contains 3 red marbles and 4 blue marbles. Two marbles are drawn at random without replacement. If the first marble drawn is red, what…Free
  2. Q30A box contains 5 green pencils and 7 yellow pencils. Two pencils are chosen at random from the box without replacement. What is the probabil…Free
  3. Q31In a sample of 40 vehicles, 18 are red, 6 are trucks, of which 2 are red. Suppose that a randomly selected vehicle is red. What is the proba…Free
  4. Q32From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when i) first card…Preview
  5. Q33$A$, $B$, and $C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\frac{3}{4…Preview
  6. Q34The probability that a student $X$ solves a problem in dynamics is $\frac{2}{5}$ and the probability that student $Y$ solves the same proble…Preview
  7. Q35$A$ speaks truth in 80% of the cases and $B$ speaks truth in 60% of the cases. Find the probability that they contradict each other in narra…Preview
  8. Q36Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result…Preview
  9. Q37Two dice are thrown together. Let $A$ be the event 'getting 6 on the first die' and $B$ be the event 'getting 2 on the second die'. Are the…Preview
  10. Q38The probability that a man who is 45 years old will be alive till he becomes 70 is $\frac{5}{12}$. The probability that his wife who is 40 y…Preview
  11. Q39A box contains 10 red balls and 15 green balls. Two balls are drawn in succession without replacement. What is the probability that, a) the…Preview
  12. Q40A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the…Preview
  13. Q41An urn contains 4 black, 5 white and 6 red balls. Two balls are drawn one after the other without replacement. What is the probability that…Preview
  14. Q42Three fair coins are tossed. What is the probability of getting three heads given that at least two coins show heads?Preview
  15. Q43Two cards are drawn one after the other from a pack of 52 cards without replacement. What is the probability that both the cards drawn are f…Preview
  16. Q44Bag $A$ contains 3 red and 2 white balls and bag $B$ contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put…Preview
  17. Q45(Activity) A bag contains 3 red and 5 white balls. Two balls are drawn at random one after the other without replacement. Find the probabili…Preview
  18. Q46A family has two children. Find the probability that both the children are girls, given that at least one of them is a girl.Preview
+Show 11 questions11 questions
  1. Q47There are three bags, each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles and Bag 3 has…Free
  2. Q48A box contains 2 blue and 3 pink balls and another box contains 4 blue and 5 pink balls. One ball is drawn at random from one of the two box…Free
  3. Q49There is a working women's hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hai…Free
  4. Q50If $E_1$ and $E_2$ are equally likely, mutually exclusive and exhaustive events and $P(A/E_1) = 0.2$, $P(A/E_2) = 0.3$. Find $P(E_1/A)$.Preview
  5. Q51Jar I contains 5 white and 7 black balls. Jar II contains 3 white and 12 black balls. A fair coin is flipped; if it is Head, a ball is drawn…Preview
  6. Q52A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probab…Preview
  7. Q53A doctor is called to see a sick child. The doctor has prior information that 80% of the sick children in that area have the flu, while the…Preview
  8. Q542% of the population have a certain blood disease of a serious form; 10% have it in a mild form; and 88% don't have it at all. A new blood t…Preview
  9. Q55A box contains three coins: two fair coins and one fake two-headed coin. A coin is picked randomly from the box and tossed. a) What is the p…Preview
  10. Q56There are three social media groups on a mobile: Group I, Group II and Group III. The probabilities that Group I, Group II and Group III sen…Preview
  11. Q57(Activity) Mr. X goes to office by Auto, Car and train. The probabilities of him travelling by these modes are $\frac{2}{7}$, $\frac{3}{7}$,…Preview
+Show 7 questions7 questions
  1. Q58If odds in favour of $X$ solving a problem are 4:3 and odds against $Y$ solving the same problem are 2:3. Find probability of: i) $X$ solvin…Free
  2. Q59The odds against John solving a problem are 4 to 3 and the odds in favor of Rafi solving the same problem are 7 to 5. What is the chance tha…Free
  3. Q60The odds against student $X$ solving a statistics problem are 8:6 and odds in favour of student $Y$ solving the same problem are 14:16. Find…Free
  4. Q61The odds against a husband who is 60 years old, living till he is 85 are 7:5. The odds against his wife who is now 56, living till she is 81…Preview
  5. Q62There are three events $A$, $B$ and $C$, one of which must, and only one can happen. The odds against the event $A$ are 7:4 and odds against…Preview
  6. Q63In a single toss of a fair die, what are the odds against the event that number 3 or 4 turns up?Preview
  7. Q64The odds in favour of $A$ winning a game of chess against $B$ are 3:2. If three games are to be played, what are the odds in favour of $A$'s…Preview