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Mathematics · Class 11 Science

Ch 12Introduction to Probability Theory — Class 11 Mathematics, concept-first.

In 1654 a gambler's dispute over how to fairly split the stakes of an interrupted game landed on the desks of two of France's leading mathematicians, Blaise Pascal and Pierre de Fermat. Their correspondence worked out, for the first time, the fundamental principles of what we now call probability theory.

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Axiomatic Probability

Classical (a priori) definition. When a random experiment's outcomes are all equally likely, the classical or a priori probability of an event is This is the everyday 'favourable over total' rule, and it silently needs t…

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Chapter contents

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12.1

Introduction

In 1654 a gambler's dispute over how to fairly split the stakes of an interrupted game landed on the desks of two of France's leading mathematicians, Blaise Pascal and Pierre de Fermat.

12.2

Basic Definitions

Before probability itself can be defined, a precise vocabulary is needed for the kind of process probability measures.

12.3

Finite Sample Space

This section restricts to sample spaces with at most a finite number of points, and builds the full vocabulary of events on top of that restricted setting.

12.4

Probability

Earlier classes introduced the frequency (a posteriori) definition of probability -- estimating from how often actually occurred across many repeated trials.

12.4.1

Classical Definition (A Priori) of Probability

Classical (a priori) definition -- Bernoulli's principle of equally likely outcomes. The basic assumption underlying the classical theory is that every outcome of the random experiment is equally like…

12.4.2

Axiomatic Approach to Probability

A.N. Kolmogorov's axiomatic system (1933). The limitations of the classical definition led to the modern definition of probability, built on the language of SETS rather than counting -- the axiomatic…

12.4.3

Odds

Odds. The word odds is used constantly in probability and statistics as another way of expressing likelihood. Odds relate the number of ways an event CAN occur to the number of ways it CANNOT.

+Exercise 12.1i10 questions
  1. Q1An experiment has the four possible mutually exclusive and exhaustive outcomes $A, B, C,$ and $D$. Check whether the following assignments o…Free
  2. Q2If two coins are tossed simultaneously, then find the probability of getting (i) one head and one tail (ii) at most two tails.Free
  3. Q3Five mangoes and 4 apples are in a box. If two fruits are chosen at random, find the probability that (i) one is a mango and the other is an…Free
  4. Q4What is the chance that (i) a non-leap year (ii) a leap year should have fifty three Sundays?Preview
  5. Q5Eight coins are tossed once, find the probability of getting (i) exactly two tails (ii) at least two tails (iii) at most two tails.Preview
  6. Q6An integer is chosen at random from the first 100 positive integers. What is the probability that the integer chosen is a prime or a multipl…Preview
  7. Q7A bag contains 7 red and 4 black balls, 3 balls are drawn at random. Find the probability that (i) all are red (ii) one red and 2 black.Preview
  8. Q8A single card is drawn from a pack of 52 cards. What is the probability that (i) the card is an ace or a king (ii) the card will be 6 or sma…Preview
  9. Q9A cricket club has 16 members, of whom only 5 can bowl. What is the probability that in a team of 11 members at least 3 bowlers are selected…Preview
  10. Q10(i) The odds that the event $A$ occurs are 5 to 7, find $P(A)$. (ii) Suppose $P(B) = \dfrac{2}{5}$. Express the odds that the event $B$ occu…Preview
12.5

Some Basic Theorems on Probability

The classical probability problems solved so far mostly involved mutually exclusive events, using the simple additivity rule .

+Exercise 12.2i6 questions
  1. Q1If $A$ and $B$ are mutually exclusive events with $P(A) = \dfrac{3}{8}$ and $P(B) = \dfrac{1}{8}$, then find (i) $P(\bar A)$ (ii) $P(A\cup B…Free
  2. Q2If $A$ and $B$ are two events associated with a random experiment for which $P(A) = 0.35$, $P(A\ \text{or}\ B) = 0.85$, and $P(A\ \text{and}…Free
  3. Q3A die is thrown twice. Let $A$ be the event, 'first die shows 5' and $B$ be the event, 'second die shows 5'. Find $P(A\cup B)$.Preview
  4. Q4The probability of an event $A$ occurring is 0.5 and $B$ occurring is 0.3. If $A$ and $B$ are mutually exclusive events, then find the proba…Preview
  5. Q5A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96. (i) What is the prob…Preview
  6. Q6The probability that a new railway bridge will get an award for its design is 0.48, the probability that it will get an award for the effici…Preview
12.6

Conditional Probability

Motivating conditional probability. Roll a fair die once, . Now ask two related but different questions.

12.6.1

Independent Events

Definition 12.15 -- independent events. Events are said to be independent if the occurrence or non-occurrence of one of them does not affect the probability of the occurrence or non-occurrence of the…

+Exercise 12.3i12 questions
  1. Q1Can two events be mutually exclusive and independent simultaneously?Free
  2. Q2If $A$ and $B$ are two events such that $P(A\cup B) = 0.7$, $P(A\cap B) = 0.2$, and $P(B) = 0.5$, then show that $A$ and $B$ are independent…Free
  3. Q3If $A$ and $B$ are two independent events such that $P(A\cup B) = 0.6$, $P(A) = 0.2$, find $P(B)$.Free
  4. Q4If $P(A) = 0.5$, $P(B) = 0.8$ and $P(B/A) = 0.8$, find $P(A/B)$ and $P(A\cup B)$.Preview
  5. Q5If for two events $A$ and $B$, $P(A) = \dfrac{3}{4}$, $P(B) = \dfrac{2}{5}$ and $A\cup B = S$ (the sample space), find the conditional proba…Preview
  6. Q6A problem in Mathematics is given to three students whose chances of solving it are $\dfrac{1}{3}, \dfrac{1}{4}$, and $\dfrac{1}{5}$. (i) Wh…Preview
  7. Q7The probability that a car being filled with petrol will also need an oil change is 0.30; the probability that it needs a new oil filter is…Preview
  8. Q8One bag contains 5 white and 3 black balls. Another bag contains 4 white and 6 black balls. If one ball is drawn from each bag, find the pro…Preview
  9. Q9Two thirds of the students in a class are boys and the rest are girls. It is known that the probability of a girl getting a first grade is 0…Preview
  10. Q10Given $P(A) = 0.4$ and $P(A\cup B) = 0.7$. Find $P(B)$ if (i) $A$ and $B$ are mutually exclusive (ii) $A$ and $B$ are independent events (ii…Preview
  11. Q11A year is selected at random. What is the probability that (i) it contains 53 Sundays (ii) it is a leap year which contains 53 Sundays?Preview
  12. Q12Suppose the chances of hitting a target by a person $X$ are 3 times in 4 shots, by $Y$ are 4 times in 5 shots, and by $Z$ are 2 times in 3 s…Preview
12.7

Total Probability of an Event

Theorem 12.10 -- Total Probability of an event. If are mutually exclusive and exhaustive events (that is, they form a partition of the sample space -- see Section 12.3), and is ANY event in , then is…

12.8

Bayes' Theorem

Thomas Bayes and the idea of reversing a conditional probability. Thomas Bayes (1702-1761) was an English statistician, philosopher, and Presbyterian minister, remembered for a specific and hugely inf…

+Exercise 12.4i5 questions
  1. Q1A factory has two Machines I and II. Machine I produces 60% of the items and Machine II produces 40% of the items of the total output. Furth…Free
  2. Q2There are two identical urns containing respectively 6 black and 4 red balls, and 2 black and 2 red balls. An urn is chosen at random and a…Free
  3. Q3A firm manufactures PVC pipes in three plants, $X$, $Y$ and $Z$. The daily production volumes from the three plants $X$, $Y$ and $Z$ are res…Preview
  4. Q4The chances of $A$, $B$ and $C$ becoming the manager of a certain company are in the ratio $5 : 3 : 2$. The probabilities that the office ca…Preview
  5. Q5An advertising executive is studying the television viewing habits of married men and women during prime time hours. Based on past viewing r…Preview
12.9

Objective Type Questions

Exercise 12.5 is the chapter's closing multiple-choice review, in the standard Samacheer Kalvi 'choose the correct or most suitable answer from the given four alternatives' format.

+Exercise 12.5i25 questions
  1. Q1Four persons are selected at random from a group of 3 men, 2 women and 4 children. The probability that exactly two of them are children is…Free
  2. Q2A number is selected from the set $\{1,2,3,\ldots,20\}$. The probability that the selected number is divisible by 3 or 4 is (1) $\dfrac{2}{5…Free
  3. Q3$A$, $B$, and $C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\dfrac{3}{…Free
  4. Q4If $A$ and $B$ are any two events, then the probability that exactly one of them occurs is (1) $P(A\cup \bar B) + P(\bar A\cup B)$ (2) $P(A\…Preview
  5. Q5Let $A$ and $B$ be two events such that $P\left(\overline{A\cup B}\right)=\dfrac{1}{6}$, $P(A\cap B)=\dfrac{1}{4}$ and $P(\overline{A})=\dfr…Preview
  6. Q6Two items are chosen from a lot containing twelve items of which four are defective. The probability that at least one of the items is defec…Preview
  7. Q7A man has 3 fifty-rupee notes, 4 hundred-rupee notes and 6 five-hundred-rupee notes in his pocket. If 2 notes are taken at random, what are…Preview
  8. Q8A 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
  9. Q9A matrix is chosen at random from the set of all $2\times2$ matrices with entries 0 or 1 only. The probability that the determinant of the m…Preview
  10. Q10A bag contains 5 white and 3 black balls. Five balls are drawn successively without replacement. The probability that they are alternately o…Preview
  11. Q11If $A$ and $B$ are two events such that $A\subset B$ and $P(B) \ne 0$, then which of the following is correct? (1) $P(A/B) = \dfrac{P(A)}{P(…Preview
  12. Q12A bag contains 6 green, 2 white, and 7 black balls. If two balls are drawn simultaneously, then the probability that both are of different c…Preview
  13. Q13If $X$ and $Y$ are two events such that $P(X/Y) = \dfrac{1}{2}$, $P(Y/X) = \dfrac{1}{3}$ and $P(X\cap Y) = \dfrac{1}{6}$, then $P(X\cup Y)$…Preview
  14. Q14An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted, and it is returned to the urn. Moreover, 2 addition…Preview
  15. Q15A number $x$ is chosen at random from the first 100 natural numbers. Let $A$ be the event of numbers which satisfy $\dfrac{(x-10)(x-50)}{x-3…Preview
  16. Q16If two events $A$ and $B$ are independent such that $P(A) = 0.35$ and $P(A\cup B) = 0.6$, then $P(B)$ is (1) $\dfrac{5}{13}$ (2) $\dfrac{1}{…Preview
  17. Q17If two events $A$ and $B$ are such that $P(A) = \dfrac{3}{10}$ and $P(A\cup B) = \dfrac{1}{2}$, then $P(\bar A\cap B)$ is (1) $\dfrac{1}{2}$…Preview
  18. Q18If $A$ and $B$ are two events such that $P(A) = P(B) = 0.4$ and $P(B/A) = 0.6$, then $P(A\cap B)$ is (1) $0.96$ (2) $0.24$ (3) $0.56$ (4) $0…Preview
  19. Q19There are three events $A$, $B$ and $C$, of which one and only one can happen. If the odds are 7 to 4 against $A$ and 5 to 3 against $B$, th…Preview
  20. Q20If $a$ and $b$ are chosen randomly from the set $\{1,2,3,4\}$ with replacement, then the probability that the equation $x^2+ax+b=0$ has real…Preview
  21. Q21It is given that events $A$ and $B$ are such that $P(A) = \dfrac{1}{4}$, $P(A/B) = \dfrac{1}{2}$ and $P(B/A) = \dfrac{2}{3}$. Then $P(B)$ is…Preview
  22. Q22In a certain college, 4% of the boys and 1% of the girls are taller than 1.8 metres. Further, 60% of the students are girls. If a student is…Preview
  23. Q23Ten coins are tossed. The probability of getting at least 8 heads is (1) $\dfrac{7}{64}$ (2) $\dfrac{7}{32}$ (3) $\dfrac{7}{16}$ (4) $\dfrac…Preview
  24. Q24The probabilities of two events $A$ and $B$ are 0.3 and 0.6 respectively. The probability that both $A$ and $B$ occur simultaneously is 0.18…Preview
  25. Q25If $m$ is a number such that $1\le m\le 5$ (i.e. $m$ is equally likely to be any one of $1,2,3,4,5$), then the probability that the quadrati…Preview
12.10

Summary

The classical probability ratio. For a sample space and event :

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 31 questions31 questions
  1. Q1If S is a sample space and A is an event and P(A) denotes the probability of A then which of the following is incorrect? (a) $P(A \cup B) =…Preview
  2. Q2If an experiment has exactly three possible mutually exclusive outcomes A, B and C, check whether the assignment of probability $P(A) = 0.51…Preview
  3. Q3'A' can hit a target 4 times in 5 shots, 'B' 3 times in 4 shots, 'C' 2 times in 3 shots, when they fire a volley. What is the chance that th…Preview
  4. Q4A number is selected from the set $\{1, 2, 3, \ldots, 20\}$. The probability that the selected number is divisible by 3 or 4 is: (a) $\dfrac…Preview
  5. Q5It is given that the events A and B are such that $P(A)=\dfrac{1}{4}$, $P(A/B)=\dfrac{1}{2}$ and $P(B/A)=\dfrac{2}{3}$. Then $P(B)$ is: (a)…Preview
  6. Q6An integer is chosen at random from the first ten positive integers. Find the probability that it is a multiple of three.Preview
  7. Q7If A and B are any two events, then the probability that exactly one of them occur is: (a) $P(A \cup \bar{B}) + P(\bar{A} \cup B)$ (b) $P(A…Preview
  8. Q8If two coins are tossed simultaneously, then find the probability of getting: (a) One head and one tail (b) Atmost two tailsPreview
  9. Q9If A and B are two events such that $P(A)=0.4$, $P(B)=0.8$ and $P(B/A)=0.6$, then $P(\overline{A} \cap B)$ is: (a) 0.56 (b) 0.96 (c) 0.66 (d…Preview
  10. Q10If X and Y be two events such that $P(X/Y)=\frac{1}{2}$, $P(Y/X)=\frac{1}{3}$ and $P(X \cap Y)=\frac{1}{6}$, then $P(X \cup Y)$ is: (a) $\fr…Preview
  11. Q11Find the total number of outcomes when 5 coins are tossed once.Preview
  12. Q12A die is rolled. If it shows an even number, then find the probability of getting 6.Preview
  13. Q13(a) There are two identical urns containing respectively 6 black and 4 red balls, 2 black and 2 red balls. An urn is chosen at random and a…Preview
  14. Q14Ten coins are tossed. The probability of getting atleast 8 heads is: (a) $\dfrac{7}{16}$ (b) $\dfrac{7}{64}$ (c) $\dfrac{7}{128}$ (d) $\dfra…Preview
  15. Q15The odds that the event A occurs is 5 to 7, then find $P(A)$.Preview
  16. Q16Suppose $P(B) = \dfrac{2}{5}$. Express the odds that the event B occurs.Preview
  17. Q17The probability of an event A occurring is 0.5 and B occurring is 0.3. If A and B are mutually exclusive events, then find $P(A\cup B)$.Preview
  18. Q18The probability of an event A occurring is 0.5 and B occurring is 0.3. If A and B are mutually exclusive events, then find $P(A\cap \bar B)$…Preview
  19. Q19The probability of an event A occurring is 0.5 and B occurring is 0.3. If A and B are mutually exclusive events, then find $P(\bar A\cap B)$…Preview
  20. Q20A factory has two machines I and II. Machine I produces 40% of items of the output and Machine II produces 60% of the items. Further 4% of i…Preview
  21. Q21If A and B are two events such that $A\subset B$ and $P(B)\neq 0$, then which of the following is correct? (a) $P(A/B)\geq P(A)$ (b) $P(A/B)…Preview
  22. Q22If two events A and B are such that $P(\bar A)=\dfrac{3}{10}$ and $P(A\cap\bar B)=\dfrac{1}{2}$ then $P(A\cap B)$ is: (a) $\dfrac{1}{4}$ (b)…Preview
  23. Q23If A and B are mutually exclusive events $P(A)=\dfrac{3}{8}$ and $P(B)=\dfrac{1}{8}$ then find $P(\bar A\cup\bar B)$.Preview
  24. Q24If A and B are any two events, then the probability that exactly one of them occurs is: (a) $P(A)+P(B)-P(A\cap B)$ (b) $P(A\cup \bar{B})+P(\…Preview
  25. Q25Four persons are selected at random from a group of 3 men, 2 women and 4 children. The probability that exactly two of them are children is:…Preview
  26. Q26An experiment has the four possible mutually exclusive and exhaustive outcomes A, B, C and D. Check if the following assignments of probabil…Preview
  27. Q27Eight coins are tossed once, find the probability of getting : (i) exactly two tails (ii) atmost two tailsPreview
  28. Q28A number is selected from the set $\{1, 2, 3, \ldots, 20\}$. The Probability that the selected number is divisible by 3 or 4 is: (a) $\dfrac…Preview
  29. Q29Ten coins are tossed. The Probability of getting at least 8 heads is: (a) $\dfrac{7}{16}$ (b) $\dfrac{7}{64}$ (c) $\dfrac{7}{128}$ (d) $\dfr…Preview
  30. Q30What is the Probability that (i) non-leap year (ii) leap year should have 53 Sundays?Preview
  31. Q31Three coins are tossed simultaneously. What is the probability of getting (i) exactly one head (ii) at least one head (iii) at most one head…Preview